#359783

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13868
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306344, #72306455, #72306473, …
maximal normal subgroup of	#72306344, #72306455, #72306473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266979);