#400816

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	#14807
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1281 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)(3,13,16,27,17,31,24,8)(4,18,20,26,21,9,28,29)(5,22,19,32,23,10,15,30);(1,3,14,16,7,17,6,24)(2,8,25,13,12,27,11,31)(4,19,28,23,21,15,20,5)(9,10,26,30,18,22,29,32);(1,4)(2,9)(3,15)(5,16)(6,20)(7,21)(8,10)(11,29)(12,18)(13,30)(14,28)(17,19)(22,27)(23,24)(25,26)(31,32);(1,5,14,19,7,23,6,15)(2,10,25,30,12,22,11,32)(3,4,16,28,17,21,24,20)(8,26,13,18,27,29,31,9)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^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	#72319908, #72319923, #72320106, …
maximal normal subgroup of	#72319908, #72319923, #72320106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308012);