#854631

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2034 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 761827);