#365331

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272527);