#720240

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
prime radical	C₂×#2407	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(6,7)(11,15)(14,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,31)(27,30)(29,32);(10,13)(14,16)(18,23)(21,24)(25,28)(30,32);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,24,23,21)(20,29,31,27)(25,30,28,32);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,31)(18,25,23,28)(19,27,26,29)(21,30,24,32);(17,21,22,24)(18,26,23,19)(20,30,31,32)(25,29,28,27)

Polycyclic presentation:

2^9|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, 627436);