#720242

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4148 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)(6,7)(9,10)(11,13)(12,15)(14,16)(18,20)(23,24);(9,11,14,12)(10,15,16,13)(17,18,22,20)(19,23,21,24);(5,6,8,7)(10,16)(11,12)(17,19,22,21)(18,23,20,24);(1,2,3,4)(5,7,8,6)(9,12,14,11)(10,15,16,13)(17,20,22,18)(19,23,21,24);(9,13,14,15)(10,11,16,12)(17,21,22,19)(18,23,20,24)

Polycyclic presentation:

2^9|b^2=f,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 627438);