#418491

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325687);