#619990

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
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

2350 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 527186);