#374980

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eh,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282176);