#480695

order	512
nilpotency class	4
isoclinism	isoclinic class of #480690
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fgi,d^c=di,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, 387891);