#2244994

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1676 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=fghi,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2152190);