#606829

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1706 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514025);