#4416319

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=f,d^2=h,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, 4323515);