#343150

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

750 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=i,d^2=h,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, 250346);