#351980

order	512
nilpotency class	3
isoclinism	isoclinic class of #348438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^c=dh,g^d=gi

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, 259176);