#707923

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,3)(7,11)(9,12)(15,19)(17,20)(21,22,26,27)(23,29,31,25)(24,32,35,28)(30,34,33,36);(13,14,16,18)(15,17,19,20)(22,27)(25,29)(28,32)(34,36);(1,2,4,3)(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,25)(24,33)(26,31)(27,29)(28,36)(30,35)(32,34);(1,3,4,2)(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,17,18,20)(21,24,26,35)(22,28,27,32)(23,30,31,33)(25,34,29,36);(21,25)(22,23)(24,34)(26,29)(27,31)(28,30)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fhi,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 615119);