#937718

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2446 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:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=fhi,d^2=g,e^2=f,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, 844914);