#676684

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2402 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

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, 583880);