#309505

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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)(9,10,12,13)(11,14,16,15)(17,18,22,27)(19,28,20,24)(21,25,31,26)(23,30,32,29);(9,11)(10,14)(12,16)(13,15)(17,19,22,20)(18,23,27,32)(21,30,31,29)(24,25,28,26);(2,5)(6,8)(10,13)(11,15)(14,16)(17,20,22,19)(18,24,27,28)(21,30,31,29)(23,25,32,26);(1,3,4,7)(2,6,5,8)(18,25)(21,31)(23,28)(24,32)(26,27)(29,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=fi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 216701);