#642922

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2545 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)(7,11)(10,12)(21,22)(23,25)(24,28)(26,27)(29,34)(30,32)(31,36)(33,35);(6,9)(10,12)(13,14,16,18)(15,17,19,20)(22,27)(23,29)(28,35)(30,31);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,25,27,34)(24,31,33,30)(28,36,35,32);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(21,24,26,33)(22,28,27,35)(23,30,29,31)(25,32,34,36);(21,25)(22,29)(23,27)(24,32)(26,34)(28,31)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=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, 550118);