#376192

order	512
nilpotency class	3
isoclinism	isoclinic class of #219491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252118, #72256282, #72256289
maximal normal subgroup of	#72252118, #72256282, #72256289

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283388);