#448673

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1002 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)(3,5)(4,6)(7,8)(17,18,23,28)(19,29,33,42)(20,25,36,41)(21,27,22,26)(24,31,38,32)(30,43,46,48)(34,40,35,39)(37,44,47,45);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,37)(26,42)(27,29)(28,38)(34,44)(35,45)(36,46)(39,48)(40,43)(41,47);(2,6)(5,8)(10,13)(14,16)(17,20,23,36)(18,25,28,41)(19,30,33,46)(21,34,22,35)(24,37,38,47)(26,39,27,40)(29,43,42,48)(31,44,32,45);(9,11,12,15)(10,14,13,16)(18,26)(19,31)(21,22)(24,38)(25,39)(27,28)(30,44)(32,33)(34,35)(37,47)(40,41)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355869);