#7576619

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	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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)(10,14)(11,12)(17,18,21,25)(19,27,30,23)(20,32,33,24)(22,26,35,39)(28,36,43,41)(29,37,31,40)(34,45,46,38)(42,48,44,47);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19,21,30)(18,23,25,27)(20,28,33,43)(22,29,35,31)(24,36,32,41)(26,40,39,37)(34,42,46,44)(38,48,45,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,28)(21,33)(22,34)(23,36)(25,32)(26,38)(27,41)(29,42)(30,43)(31,44)(35,46)(37,47)(39,45)(40,48);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(18,25)(19,29)(22,35)(23,37)(24,32)(27,40)(28,42)(30,31)(34,46)(36,47)(41,48)(43,44);(2,8)(6,7)(9,12,13,11)(10,15,14,16)(18,25)(20,33)(23,27)(26,39)(28,43)(34,46)(37,40)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=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, 7483815);