#615573

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2859 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522769);