#235984

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

1398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,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,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, 143180);