#5956383

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2556 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=gi,e^2=g,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, 5863579);