#4416334

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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

2114 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)(5,6,9,10)(7,11,12,8)(15,18)(16,20)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34);(5,7)(6,8)(9,12)(10,11)(13,14,17,19)(15,16,18,20)(23,29)(27,30)(31,35)(34,36);(1,2,3,4)(7,12)(8,11)(13,15,17,18)(14,16,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,6)(7,8)(9,10)(11,12)(13,16)(14,18)(15,19)(17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,8)(6,7)(9,11)(10,12)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323530);