#419807

order	512
nilpotency class	3
isoclinism	isoclinic class of #419807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15273
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1602 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215958, #72216192, #72216203, …
maximal normal subgroup of	#72215958, #72216192, #72216203, …
maximal quotient of	#72237180, C₂×#419807, #72255712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327003);