#234041

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
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)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(21,22,27,31)(23,32,29,35)(24,34,28,33)(25,30,26,36);(1,2)(3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20)(21,23)(22,28)(24,31)(25,34)(26,33)(27,29)(30,35)(32,36);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19)(21,24,27,28)(22,29,31,23)(25,32,26,35)(30,33,36,34);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12)(21,25,27,26)(22,30,31,36)(23,33,29,34)(24,35,28,32)

Polycyclic presentation:

2^9|a^2=i,c^2=i,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,d^c=dh,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, 141237);