#235665

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#3106	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1440 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)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22)(23,32)(24,29)(25,30)(26,31)(27,34)(28,33)(35,36);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(21,23,26,35)(22,27,31,28)(24,33,25,34)(29,32,30,36);(2,4)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,36)(30,32);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24,26,25)(22,29,31,30)(23,33,35,34)(27,32,28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142861);