#1296112

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

1895 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)(7,12)(8,10)(14,17)(15,19)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,35,36,33);(5,6,9,11)(7,10,12,8)(14,17)(18,20)(22,27)(23,30)(29,32)(31,36);(5,7)(6,10)(8,11)(9,12)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,33)(32,35)(34,36);(1,2,3,4)(5,8,9,10)(6,7,11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,36,32,31)

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fh,d^2=fhi,e^2=f,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, 1203308);