#248875

order	512
nilpotency class	3
isoclinism	isoclinic class of #248839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156071);