#405945

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14921
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317787, #72317872, #72318474, …
maximal normal subgroup of	#72317787, #72317872, #72318474, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313141);