#397952

order	512
nilpotency class	3
isoclinism	isoclinic class of #397952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14757
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 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)(3,11)(4,16)(5,20)(6,10)(7,14)(8,18)(9,21)(12,27)(13,22)(15,26)(17,24)(19,23)(25,31)(28,30)(29,32);(1,3,5,14,6,15,21,27)(2,7,9,11,10,12,20,26)(4,13,17,29,19,30,18,31)(8,22,23,25,24,28,16,32);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,26,27)(13,28,30,22)(16,18)(17,23)(19,24)(20,21)(25,29,32,31);(1,4)(2,8)(3,13)(5,17)(6,19)(7,22)(9,23)(10,24)(11,25)(12,28)(14,29)(15,30)(16,20)(18,21)(26,32)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316368, C₂×#397952
maximal normal subgroup of	#72316368, C₂×#397952
maximal quotient of	C₂×#397952, #72316830

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305148);