#246738

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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

1144 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=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, 153934);