#759012

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=fi,e^2=fhi,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, 666208);