#3793132

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2317 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fi,d^2=fi,e^2=fgi,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, 3700328);