#3474108

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1672 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)(5,6,9,10)(7,11,12,8)(13,14,17,16)(15,18,19,20)(21,22,26,29)(23,30,24,27)(25,28,34,33)(31,36,32,35);(5,7)(6,8)(9,12)(10,11)(15,19)(18,20)(21,23,26,24)(22,27,29,30)(25,31,34,32)(28,35,33,36);(6,10)(8,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,23)(22,27,29,30)(25,31,34,32)(28,36,33,35);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,35)(32,36);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3381304);