#307614

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 214810);