#1609999

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1828 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1517195);