#7205067

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1476 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)(11,14)(12,15)(13,16)(17,18,22,27)(19,29,32,24)(20,26,34,42)(21,28,23,25)(30,43,45,38)(31,39,35,37)(33,40,36,41)(44,47,46,48);(1,3)(2,5)(4,7)(6,8)(9,11,14,10)(12,15,13,16)(17,19)(18,24)(20,30)(21,35)(22,32)(23,31)(25,37)(26,38)(27,29)(28,39)(33,46)(34,45)(36,44)(40,47)(41,48)(42,43);(3,7)(5,8)(9,12,14,13)(10,15,11,16)(18,25)(21,23)(24,37)(26,40)(27,28)(29,39)(31,35)(33,36)(38,47)(41,42)(43,48)(44,46);(2,6)(3,7)(9,13,14,12)(10,15,11,16)(17,20,22,34)(18,26,27,42)(19,30,32,45)(21,36,23,33)(24,38,29,43)(25,40,28,41)(31,46,35,44)(37,47,39,48);(3,7)(5,8)(10,11)(15,16)(17,21,22,23)(18,25,27,28)(19,31,32,35)(20,33,34,36)(24,39,29,37)(26,41,42,40)(30,44,45,46)(38,47,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,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=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, 7112263);