#1294884

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1838 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)(6,8)(7,11)(13,14,18,16)(15,19,17,20)(21,22)(23,29)(24,30)(25,33)(26,27)(28,35)(31,32)(34,36);(13,15,18,17)(14,19,16,20)(22,27)(23,30)(28,33)(31,36);(5,6,9,8)(7,10,11,12)(13,16,18,14)(15,19,17,20)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,34)(35,36);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,19,16,20)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,33)(32,35);(5,8,9,6)(7,12,11,10)(21,25,26,35)(22,28,27,33)(23,31,30,36)(24,32,29,34)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1202080);