#399217

order	512
nilpotency class	3
isoclinism	isoclinic class of #393998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14777
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1344 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306453, #72306456, #72306473, …
maximal normal subgroup of	#72306453, #72306456, #72306473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306413);