#236229

order	512
nilpotency class	3
isoclinism	isoclinic class of #233958
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	C₂×#3212	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1345 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)(23,32)(24,29)(25,30)(26,31)(27,34)(28,33)(35,36);(1,2)(3,4)(5,7)(6,10)(8,17)(9,14)(11,16)(12,20)(13,19)(15,18)(21,23,26,35)(22,27,31,28)(24,33,25,34)(29,32,30,36);(2,4)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,36)(30,32);(7,14)(10,19)(11,20)(15,17)(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=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,f^b=fi,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143425);