#6345404

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306822, #72306828, #72306874, …
maximal normal subgroup of	#72306822, #72306828, #72306874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252600);