#383760

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,h^c=hi,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, 290956);