#381175

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,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, 288371);