#7582419

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	#24129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7489615);