#7484513

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

1382 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)(17,18,23,25,22,27,21,28)(19,24,36,41,29,40,31,38)(20,32,37,44,34,26,35,42)(30,43,46,47,33,39,45,48);(1,2)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(17,19)(18,24)(20,33)(21,31)(22,29)(23,36)(25,41)(26,43)(27,40)(28,38)(30,34)(32,39)(35,46)(37,45)(42,47)(44,48);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,29)(20,34)(21,23)(24,38)(26,44)(27,28)(32,42)(39,48)(40,41)(43,47)(45,46);(3,8)(6,7)(10,13)(14,15)(17,20)(18,26)(19,30)(21,35)(22,34)(23,37)(24,39)(25,42)(27,32)(28,44)(29,33)(31,45)(36,46)(38,47)(40,43)(41,48);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(18,27)(20,34)(24,40)(25,28)(30,33)(35,37)(38,41)(45,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,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, 7391709);