#626528

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3232 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)(6,8)(11,15)(14,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28);(10,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(1,2,3,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32);(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 533724);