#7208288

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	#53065	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,28,40)(20,31,32,24)(21,34,35,25)(23,36,37,30)(29,43,42,41)(33,39,47,46)(38,44,45,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(19,28)(20,32)(23,37)(24,31)(27,40)(30,36)(33,47)(39,46);(11,15)(13,16)(17,19,21,30,22,28,35,36)(18,23,25,27,26,37,34,40)(20,29,33,44,32,42,47,48)(24,38,39,41,31,45,46,43);(3,8)(5,7)(10,14)(11,15)(17,20,22,32)(18,24,26,31)(19,29,28,42)(21,33,35,47)(23,38,37,45)(25,39,34,46)(27,41,40,43)(30,44,36,48);(2,6)(5,7)(11,15)(13,16)(20,32)(24,31)(29,42)(33,47)(38,45)(39,46)(41,43)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7115484);