#649094

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=h,d^2=f,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, 556290);