#4378525

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

2428 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,29)(25,30)(27,32)(28,33)(31,34)(35,36);(15,19)(17,20)(22,23)(25,33)(27,29)(31,36);(1,2)(3,4)(13,15)(14,17)(16,19)(18,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7,9,12)(6,10,8,11)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(5,8,9,6)(7,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|a^2=hi,c^2=f,d^2=i,e^2=fi,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, 4285721);