#351858

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,g^d=gi

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, 259054);