#4664679

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1777 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)(5,6,11,10)(7,8,12,9)(15,19)(18,20)(21,22,26,30)(23,25,29,27)(24,32,35,28)(31,36,33,34);(5,7)(6,9)(8,10)(11,12)(14,17)(18,20)(21,23,26,29)(22,27,30,25)(24,33,35,31)(28,34,32,36);(5,8,11,9)(6,12,10,7)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,36)(32,34)(33,35);(1,2,3,4)(5,9,11,8)(6,12,10,7)(21,24)(22,28)(23,31)(25,34)(26,35)(27,36)(29,33)(30,32);(5,10,11,6)(7,8,12,9)(13,15)(14,18)(16,19)(17,20)(21,25,26,27)(22,29,30,23)(24,34,35,36)(28,33,32,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4571875);