#615589

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2968 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fi,d^2=g,e^2=f,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, 522785);