#1200417

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1932 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:

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

Polycyclic presentation:

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

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, 1107613);