#234105

order	512
nilpotency class	3
isoclinism	isoclinic class of #234099
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

1233 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)(11,16)(15,19)(17,20)(21,22,25,29)(23,26,30,35)(24,32,34,28)(27,33,36,31);(1,2)(3,4)(5,7,10,17)(6,11,13,19)(8,16,9,15)(12,20,18,14)(21,23)(22,26)(24,31)(25,30)(27,32)(28,36)(29,35)(33,34);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19)(21,23)(22,27)(24,33)(25,30)(26,32)(28,35)(29,36)(31,34);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(23,30)(26,35)(27,36)(31,33)

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=dhi,e^b=eh,d^c=dhi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141301);