#351920

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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)(5,6,9,13)(7,10,14,17)(8,11,16,12)(15,18,20,19)(22,26)(23,30)(24,27)(28,33)(31,32)(34,36);(5,7)(6,10)(8,15)(9,14)(11,18)(12,19)(13,17)(16,20)(21,22,27,33,25,28,24,26)(23,29,30,36,31,35,32,34);(1,2,3,4)(7,14)(10,17)(15,20)(18,19)(22,28)(23,31)(26,33)(30,32);(6,11)(8,16)(10,18)(12,13)(15,20)(17,19)(21,23)(22,29)(24,32)(25,31)(26,34)(27,30)(28,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259116);