#7203013

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,23,27)(19,28,29,42)(20,33,35,25)(21,26,22,37)(24,31,38,32)(30,45,44,43)(34,41,36,40)(39,46,48,47);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(19,29)(24,38)(28,42)(30,44)(31,32)(39,48)(43,45)(46,47);(3,7)(5,8)(10,11)(14,15)(17,19,23,29)(18,24,27,38)(20,30,35,44)(21,32,22,31)(25,39,33,48)(26,42,37,28)(34,47,36,46)(40,43,41,45);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20,23,35)(18,25,27,33)(19,30,29,44)(21,36,22,34)(24,39,38,48)(26,41,37,40)(28,43,42,45)(31,47,32,46);(3,7)(5,8)(10,11)(14,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(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=i,d^2=fi,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=dfhi,e^b=ehi,e^c=ei,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, 7110209);