#517955

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2128 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fgh,d^2=i,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425151);