#376176

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
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

1562 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:

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

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=dei,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

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256282, #72256284, #72256293
maximal normal subgroup of	#72256282, #72256284, #72256293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283372);