#6386118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293314);