#7544695

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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)(3,8)(4,6)(5,7)(9,10,13,11)(12,15,16,14)(17,18)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,39)(29,40)(31,45)(33,42)(35,46)(36,44)(37,41)(38,43)(47,48);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,33)(21,30)(22,37)(24,42)(25,27)(26,45)(28,34)(29,35)(31,36)(32,39)(38,48)(40,43)(41,44)(46,47);(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,28)(21,34)(22,35)(23,39)(25,32)(26,43)(27,42)(29,31)(30,33)(36,38)(37,48)(40,41)(44,46)(45,47);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20)(18,24)(19,29)(21,34)(22,38)(23,40)(25,32)(26,46)(27,47)(28,31)(30,48)(33,37)(35,36)(39,41)(42,45)(43,44);(2,7)(6,8)(18,25)(20,34)(23,27)(26,44)(28,33)(29,48)(35,38)(41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7451891);