#1397808

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1986 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)(13,14,17,15)(16,19,20,18)(21,22)(23,30)(24,28)(25,33)(26,27)(29,35)(31,32)(34,36);(6,9)(10,12)(13,15,17,14)(16,19,20,18)(22,27)(23,28)(29,33)(31,36);(5,6,8,9)(7,10,11,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,24)(25,31)(26,28)(27,30)(29,32)(33,34)(35,36);(1,2)(3,4)(21,24)(22,28)(23,27)(25,34)(26,30)(29,31)(32,35)(33,36);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)(21,25,26,35)(22,29,27,33)(23,31,28,36)(24,32,30,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fh,e^2=h,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, 1305004);