#7202619

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(19,29)(20,26)(21,28)(22,24)(23,27)(25,33)(30,43)(31,37)(32,40)(34,38)(35,42)(36,41)(39,45)(44,48)(46,47);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(18,24)(20,34)(25,37)(27,28)(29,40)(30,44)(35,36)(45,46);(3,7)(5,8)(10,11)(14,15)(17,19,22,32)(18,25,24,37)(20,30,34,44)(21,33,23,31)(26,39,38,47)(27,29,28,40)(35,46,36,45)(41,48,42,43);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20,22,34)(18,26,24,38)(19,30,32,44)(21,36,23,35)(25,39,37,47)(27,42,28,41)(29,43,40,48)(31,46,33,45);(3,7)(5,8)(10,11)(14,15)(17,21,22,23)(18,27,24,28)(19,31,32,33)(20,35,34,36)(25,40,37,29)(26,41,38,42)(30,45,44,46)(39,43,47,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,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=eh,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, 7109815);