#7208293

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(17,18,23,27)(19,28,29,42)(20,33,34,25)(21,26,22,37)(24,31,38,32)(30,45,44,43)(35,41,36,40)(39,46,48,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,34)(24,38)(25,33)(28,42)(31,32)(35,36)(40,41);(11,15)(13,16)(17,19,21,31,23,29,22,32)(18,24,37,42,27,38,26,28)(20,30,36,47,34,44,35,46)(25,39,40,45,33,48,41,43);(3,8)(5,7)(10,14)(11,15)(17,20,23,34)(18,25,27,33)(19,30,29,44)(21,36,22,35)(24,39,38,48)(26,41,37,40)(28,43,42,45)(31,47,32,46);(2,6)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,35,34,36)(24,28,38,42)(25,40,33,41)(30,46,44,47)(39,45,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^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, 7115489);