#946768

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,d^2=i,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, 853964);