#401055

order	512
nilpotency class	3
isoclinism	isoclinic class of #393998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14811
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1675 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:

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

Polycyclic presentation:

2^9|a^2=f,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	#72318900, #72318940, #72318991, …
maximal normal subgroup of	#72318900, #72318940, #72318991, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308251);