#149604

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=ci,d^b=di,e^b=efi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56800);