#719405

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,d^2=fi,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, 626601);