#418515

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1770 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)(7,17)(8,21)(9,12)(10,15)(11,16)(13,19)(14,22)(18,32)(20,26)(23,30)(24,27)(25,29)(28,34)(31,33)(35,36);(6,12)(8,22)(9,10)(13,17)(15,16)(18,33)(19,25)(21,27)(23,24)(26,29)(28,32)(34,35);(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,26)(14,28)(15,17)(16,29)(21,31)(22,33)(23,35)(24,34)(27,36)(30,32);(2,4)(5,8,10,23,11,24,9,22)(6,14,12,27,16,30,15,21)(7,18,25,35,20,34,19,33)(13,28,26,36,29,32,17,31)

Polycyclic presentation:

2^9|d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=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, 325711);