#333296

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

763 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

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, 240492);