#2241778

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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

1508 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=fi,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, 2148974);