#612421

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519617);