#4435100

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4342296);