#419827

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1601 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)(7,17,19,12)(8,13)(9,26)(10,16)(11,15)(14,21,32,20)(18,27,35,34)(22,29,36,28)(23,33)(24,31)(25,30);(1,2)(3,4)(5,7)(6,12)(8,22)(9,21)(10,20)(11,19)(13,28)(14,26)(15,17)(16,32)(18,23)(24,35)(25,36)(27,31)(29,30)(33,34);(2,4)(5,8,11,25)(6,13,15,30)(7,18,19,35)(9,24,10,23)(12,27,17,34)(14,29,32,28)(16,33,26,31)(20,36,21,22);(5,7)(6,14,15,32)(8,18)(9,20)(10,21)(11,19)(12,16,17,26)(13,29,30,28)(22,23)(24,36)(25,35)(27,33,34,31)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,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, 327023);