#5956350

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=fi,e^2=f,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, 5863546);