#4310188

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4217384);