#7529214

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=i,e^2=fi,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, 7436410);