#236078

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1217 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)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(21,22,27,30)(23,31,34,35)(24,33,36,32)(25,28,26,29);(1,2)(3,4)(5,7,10,16)(6,11,14,20)(8,15,9,17)(12,13,19,18)(21,23,27,34)(22,24,30,36)(25,33,26,32)(28,31,29,35);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)(21,24)(22,23)(25,35)(26,31)(27,36)(28,32)(29,33)(30,34);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(21,25,27,26)(22,28,30,29)(23,32,34,33)(24,35,36,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 143274);