#7206251

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	#53073	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1568 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,31)(21,23)(22,35)(24,30)(25,32)(26,34)(28,42)(29,40)(33,39)(36,37)(38,48)(41,43)(44,45)(46,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(18,23)(24,37)(25,31)(26,35)(27,40)(38,45)(39,46)(41,42);(11,15)(13,16)(17,19,21,29)(18,24,23,37)(20,28,32,43)(22,36,34,30)(25,38,31,45)(26,40,35,27)(33,48,47,44)(39,42,46,41);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20,21,32)(18,25,23,31)(19,28,29,43)(22,33,34,47)(24,38,37,45)(26,39,35,46)(27,41,40,42)(30,44,36,48);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(20,32)(25,31)(28,43)(33,47)(38,45)(39,46)(41,42)(44,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7113447);