#480045

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

962 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,14,20,15,21,35,48,16,22,36,49,13,19,33,46)(11,23,26,51,25,40,55,47,27,45,34,61,32,17,37,39)(12,18,30,43,31,44,60,69,28,41,56,65,29,42,58,67)(24,50,54,70,53,63,71,68,52,66,59,72,57,38,62,64);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,24)(13,32)(14,34)(15,25)(16,27)(18,38)(19,45)(20,47)(21,23)(22,40)(26,36)(28,52)(29,57)(30,59)(31,53)(33,55)(35,37)(39,49)(41,63)(42,66)(43,68)(44,50)(46,51)(48,61)(54,56)(58,71)(60,62)(64,65)(67,70)(69,72);(9,12)(10,18)(11,24)(13,29)(14,30)(15,31)(16,28)(17,38)(19,42)(20,43)(21,44)(22,41)(23,50)(25,53)(26,54)(27,52)(32,57)(33,58)(34,59)(35,60)(36,56)(37,62)(39,64)(40,63)(45,66)(46,67)(47,68)(48,69)(49,65)(51,70)(55,71)(61,72);(12,28)(18,41)(24,52)(29,31)(30,56)(38,63)(42,44)(43,65)(50,66)(53,57)(54,59)(58,60)(62,71)(64,68)(67,69)(70,72)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=ei,f^b=fgi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387241);