#150090

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	#121	C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

210 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=fi,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,c^b=ci,d^b=dfi,e^b=ei

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, 57286);