#1040823

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fhi,d^2=g,e^2=fg,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, 948019);