#315332

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efi,c^2=hi,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222528);