#916019

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2426 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)(11,15)(14,16)(18,20)(22,23)(26,29)(28,31);(1,2,6,5)(3,8,4,7)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,31,32,28);(1,3,6,4)(2,7,5,8)(9,10,13,12)(11,14,15,16)(17,18,21,20)(19,22,24,23)(25,27,30,32)(26,31,29,28);(1,4,6,3)(2,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,32)(30,31);(1,5,6,2)(3,8,4,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,28,32,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=ghi,d^2=g,e^2=ghi,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, 823215);