#479909

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,27)(22,30)(23,28)(24,29)(31,35)(32,36)(33,37)(34,38)(39,40);(2,5)(6,8)(9,11,12,20,13,21,22,33,14,19,23,31,24,32,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224814, #72224821, #72224822, …
maximal normal subgroup of	#72224814, #72224821, #72224822, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387105);