#5956465

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 33554432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3718 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863661);