#390399

order	512
nilpotency class	3
isoclinism	isoclinic class of #31282
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 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)(6,12,15,24)(7,16,19,25)(14,20,30,27)(18,33,35,31);(5,6)(7,17)(8,20)(9,24)(10,12)(11,15)(13,19)(14,21)(16,26)(18,34)(22,30)(23,27)(25,29)(28,31)(32,33)(35,36);(2,4)(5,7)(6,13,15,17)(8,18)(9,16)(10,25)(11,19)(12,26,24,29)(14,28,30,32)(20,36,27,34)(21,33)(22,31)(23,35);(2,4)(5,8,11,23)(6,14,15,30)(7,18,19,35)(9,22,10,21)(12,27,24,20)(13,28,17,32)(16,31,25,33)(26,34,29,36)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 297595);