#1397987

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2119 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)(6,9)(7,11)(22,24)(27,31)(28,32)(34,35);(14,17)(18,20)(21,22,26,24)(23,27,30,31)(25,28,33,32)(29,34,36,35);(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(21,24,26,22)(23,27,30,31)(25,28,33,32)(29,35,36,34);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,25,26,33)(22,28,24,32)(23,29,30,36)(27,34,31,35)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,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, 1305183);