#308931

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=fi,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,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216127);