#395251

order	512
nilpotency class	3
isoclinism	isoclinic class of #395250
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14722
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1746 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)(4,9)(5,13)(6,14)(8,11)(10,19)(12,17)(15,28)(16,23)(18,24)(20,27)(21,25)(22,31)(26,30)(29,32);(1,3,7,2,8,14,6,11)(4,9,17,23,18,24,16,12)(5,13,21,10,20,27,19,25)(15,22,31,28,30,32,29,26);(1,4)(2,9)(3,12)(5,15)(6,16)(7,17)(8,18)(10,22)(11,24)(13,26)(14,23)(19,29)(20,30)(21,31)(25,32)(27,28);(1,5,6,19,8,20,7,21)(2,10,3,13,11,25,14,27)(4,15,16,29,18,30,17,31)(9,22,12,26,24,32,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215936, #72215972, #72215995, …
maximal normal subgroup of	#72215936, #72215972, #72215995, …
maximal quotient of	C₂×#395251, #72252126, #72252142, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302447);