#6374378

order	512
nilpotency class	3
isoclinism	isoclinic class of #60753
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

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, 6281574);