#7548618

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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,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, 7455814);