#482947

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,22,28,23,27,19,24)(20,25,31,39,33,41,29,37)(21,26,35,44,36,43,30,38)(32,40,46,48,34,42,45,47);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(17,19,23,22)(18,24)(20,31,33,29)(21,30,36,35)(25,39)(26,38)(27,28)(32,46,34,45)(37,41)(40,48)(42,47)(43,44);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,20,23,33)(18,25,27,41)(19,29,22,31)(21,34,36,32)(24,37,28,39)(26,42,43,40)(30,46,35,45)(38,48,44,47);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,21,23,36)(18,26,27,43)(19,30,22,35)(20,32,33,34)(24,38,28,44)(25,40,41,42)(29,45,31,46)(37,47,39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,d^c=dh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390143);