#4319978

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 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)(5,6,8,11)(7,10,12,9)(13,14,18,17)(15,20,16,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34);(5,7)(6,9)(8,12)(10,11)(13,14,18,17)(15,19,16,20)(23,29)(27,30)(31,35)(34,36);(7,12)(9,10)(13,15,18,16)(14,19,17,20)(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)(13,16,18,15)(14,20,17,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,6)(7,10)(8,11)(9,12)(13,17,18,14)(15,20,16,19)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4227174);