#946637

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

2413 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)(6,9)(7,11)(13,14,17,19)(15,20,18,16)(22,24)(25,33)(27,31)(29,35);(7,11)(10,12)(15,18)(16,20)(21,22,26,24)(23,27,30,31)(25,28,33,32)(29,34,35,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,16,19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,24,26,22)(23,27,30,31)(25,32,33,28)(29,34,35,36);(21,25)(22,28)(23,29)(24,32)(26,33)(27,34)(30,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,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, 853833);