#234048

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
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	C₂×C₄²	C₂⁴
prime radical	#37621	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1373 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)(7,11)(8,16)(9,13)(10,15)(12,18)(14,20)(17,19)(21,22,25,29)(23,26,30,34)(24,27,32,28)(31,33,36,35);(1,2)(3,4)(5,7)(6,10)(8,17)(9,14)(11,16)(12,20)(13,19)(15,18)(21,23)(22,26)(24,31)(25,30)(27,33)(28,35)(29,34)(32,36);(2,4)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(7,14)(10,19)(11,20)(15,17)(23,30)(26,34)(31,36)(33,35)

Polycyclic presentation:

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

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, 141244);