#1075197

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1812 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=h,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 982393);