#4329145

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1934 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=fg,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, 4236341);