#1295654

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2301 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)(5,6,10,8)(7,12,9,11)(13,14)(15,19)(16,18)(17,20)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35);(5,6,10,8)(7,11,9,12)(13,15,18,17)(14,19,16,20)(22,27)(23,28)(29,35)(30,36);(5,7,10,9)(6,11,8,12)(13,16,18,14)(15,19,17,20)(21,23,26,28)(22,24,27,32)(25,30,34,36)(29,31,35,33);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24)(22,28)(23,27)(25,33)(26,32)(29,30)(31,34)(35,36);(5,9,10,7)(6,12,8,11)(13,17,18,15)(14,19,16,20)(21,25)(22,29)(23,30)(24,31)(26,34)(27,35)(28,36)(32,33)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,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, 1202850);