#318079

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

645 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)(5,6,10,14)(7,12,15,20)(8,17,16,11)(9,19,18,13)(21,22)(23,29)(24,27)(25,28)(26,32)(30,34)(31,36)(33,35);(5,7,9,16,10,15,18,8)(6,11,13,12,14,17,19,20)(21,23,30,31,24,32,25,33)(22,26,34,35,27,29,28,36);(2,4)(7,15)(8,16)(11,17)(12,20)(23,31)(25,30)(26,35)(28,34)(29,36)(32,33);(5,8,18,15,10,16,9,7)(6,12,19,11,14,20,13,17)(22,27)(26,29)(28,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225275);