#4935919

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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

2124 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,19,20,17)(22,26)(23,29)(24,32)(34,36);(6,9)(10,11)(22,26)(27,30)(28,33)(34,36);(5,7,8,12)(6,10,9,11)(13,15,16,20)(14,17,18,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2)(3,4)(13,14)(15,19)(16,18)(17,20)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,6)(7,10)(8,9)(11,12)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4843115);