#376175

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256289, #72256291, #72256293
maximal normal subgroup of	#72256289, #72256291, #72256293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283371);