#7484190

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

1377 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,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, 7391386);