#310137

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

715 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)(9,10,11,14,12,15,16,13)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(10,13)(11,16)(14,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,30,32,31);(2,5)(6,8)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,7)(2,6,5,8)(10,13)(11,16)(14,15)(18,23)(22,25)(24,32)(30,31)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=f,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, 217333);