#4344069

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

1848 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)(7,11)(8,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,27)(23,30,31,28)(24,25,35,29)(32,33,34,36);(5,6,10,9)(7,8,11,12)(13,15,17,20)(14,18,19,16)(22,27)(23,31)(25,29)(32,34);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,31)(22,28,27,30)(24,34,35,32)(25,33,29,36);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16)(14,15)(17,18)(19,20)(21,24)(22,29)(23,32)(25,27)(26,35)(28,33)(30,36)(31,34);(5,9,10,6)(7,12,11,8)(13,16)(14,15)(17,18)(19,20)(21,25,26,29)(22,24,27,35)(23,33,31,36)(28,34,30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=hi,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, 4251265);