#2244974

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=ghi,d^2=g,e^2=h,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, 2152170);