#343155

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24117	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=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, 250351);