#7552613

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7459809);