#4423105

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2764 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=h,d^2=g,e^2=fh,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, 4330301);