#236182

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1258 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)(9,18)(10,13)(11,15)(16,20)(17,19)(21,22,27,32)(23,33,29,36)(24,35,28,34)(25,30,26,31);(1,2)(3,4)(5,7,10,17)(6,11,13,20)(8,15,9,16)(12,19,18,14)(21,23,27,29)(22,28,32,24)(25,34,26,35)(30,33,31,36);(2,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,20,14)(21,24,27,28)(22,29,32,23)(25,33,26,36)(30,34,31,35);(5,8,10,9)(6,12,13,18)(7,16,17,15)(11,14,20,19)(21,25,27,26)(22,30,32,31)(23,34,29,35)(24,36,28,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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=dhi,e^b=eh,f^b=fi,d^c=di,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, 143378);