#7481452

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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)(17,18,22,27)(19,23,30,38)(20,31,32,25)(21,24,34,26)(28,41,42,36)(29,35,44,37)(33,40,45,39)(43,47,48,46);(2,5)(3,7)(9,10,12,13)(11,15,16,14)(17,19,22,30)(18,23,27,38)(20,28,32,42)(21,29,34,44)(24,35,26,37)(25,36,31,41)(33,43,45,48)(39,46,40,47);(9,10)(11,14)(12,13)(15,16)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(31,40)(33,45)(36,46)(37,38)(41,47)(43,48);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,22,32)(18,25,27,31)(19,28,30,42)(21,33,34,45)(23,36,38,41)(24,39,26,40)(29,43,44,48)(35,46,37,47);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(20,32)(25,31)(28,42)(33,45)(36,41)(39,40)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7388648);