#310070

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

646 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)(3,6)(4,5)(7,8)(17,18,23,26,22,28,21,27)(19,25,31,24,20,30,29,32);(10,13)(11,15)(14,16)(17,19,22,20)(18,24,28,32)(21,31,23,29)(25,27,30,26);(2,5)(6,8)(9,10,12,14)(11,16,15,13)(17,20,22,19)(18,25,28,30)(21,31,23,29)(24,27,32,26);(1,3,4,7)(2,6,5,8)(18,26)(21,23)(24,25)(27,28)(29,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217266);