#394344

order	512
nilpotency class	3
isoclinism	isoclinic class of #394343
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14716
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241171, #72241179, #72241305, …
maximal normal subgroup of	#72241171, #72241179, #72241305, …
maximal quotient of	C₂×#394344, #72316842, #72316938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301540);