#624438

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

2592 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 531634);