#419825

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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1671 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)(7,16,19,12)(8,21,22,36)(9,25)(10,15)(11,14)(13,24,30,23)(17,26,20,28)(18,32)(27,31)(29,33)(34,35);(1,2)(3,4)(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,29)(14,16)(15,26)(21,35)(22,34)(23,31)(24,33)(25,28)(27,30)(32,36);(2,4)(7,17)(9,10)(12,26)(15,25)(16,28)(18,31)(19,20)(21,36)(23,24)(27,32)(29,35)(33,34);(5,8)(6,13,14,30)(7,18)(9,23)(10,24)(11,22)(12,27,16,29)(15,21,25,36)(17,31)(19,34)(20,33)(26,32,28,35)

Polycyclic presentation:

2^9|a^2=f,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.

Relationships

maximal subgroup of	#72252199, #72256282, #72256291
maximal normal subgroup of	#72252199, #72256282, #72256291

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327021);