#610159

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

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, 517355);