#4452831

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2186 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)(6,8)(7,11)(14,18)(15,20)(21,22,25,29)(23,27,31,26)(24,28,32,35)(30,34,36,33);(5,6,9,8)(7,12,11,10)(13,14)(15,16)(17,18)(19,20)(21,23)(22,26)(24,30)(25,31)(27,29)(28,33)(32,36)(34,35);(5,7,9,11)(6,10,8,12)(13,15,17,20)(14,16,18,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,29)(27,31)(30,33)(32,35)(34,36);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,15,18,20)(21,24,25,32)(22,28,29,35)(23,30,31,36)(26,33,27,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4360027);