#7484925

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=hi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7392121);