#7484194

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	#53152	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(19,24,34,36,31,38,30,39)(20,26,35,40,32,42,33,41)(29,37,45,46,43,48,44,47);(1,2)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(17,19,22,31)(18,24,27,38)(20,29,32,43)(21,30,23,34)(25,36,28,39)(26,37,42,48)(33,44,35,45)(40,46,41,47);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(18,25)(21,23)(24,36)(26,40)(27,28)(30,34)(33,35)(37,46)(38,39)(41,42)(44,45)(47,48);(3,8)(6,7)(10,13)(14,15)(17,20,22,32)(18,26,27,42)(19,29,31,43)(21,33,23,35)(24,37,38,48)(25,40,28,41)(30,44,34,45)(36,46,39,47);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(18,27)(20,32)(24,38)(25,28)(29,43)(33,35)(36,39)(44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=f,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, 7391390);