#946770

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

2106 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)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35);(7,11)(10,12)(13,15,17,20)(14,18,16,19)(22,23)(27,29)(28,30)(34,35);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,18)(17,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,18,20,19)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35);(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,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, 853966);