#946760

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

1938 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)(5,6,8,10)(7,11,12,9)(13,14,18,15)(16,20,17,19)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35);(7,12)(9,11)(13,15,18,14)(16,19,17,20)(22,26)(27,29)(28,32)(34,35);(5,6)(7,9)(8,10)(11,12)(13,16,18,17)(14,19,15,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,17,18,16)(14,20,15,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=fi,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, 853956);