#7203008

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(17,18,21,26)(19,28,30,23)(20,25,32,41)(22,35,34,27)(24,36,40,33)(29,43,45,38)(31,39,47,44)(37,46,42,48);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19,21,30)(18,23,26,28)(20,29,32,45)(22,31,34,47)(24,37,40,42)(25,38,41,43)(27,39,35,44)(33,46,36,48);(3,7)(5,8)(10,11)(14,15)(17,20)(18,24)(19,29)(21,32)(22,36)(23,37)(25,35)(26,40)(27,41)(28,42)(30,45)(31,48)(33,34)(38,44)(39,43)(46,47);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,38)(24,35)(26,41)(27,40)(28,43)(30,45)(31,46)(34,36)(37,44)(39,42)(47,48);(3,7)(5,8)(10,11)(14,15)(18,26)(19,30)(20,32)(27,35)(31,47)(33,36)(37,42)(38,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,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, 7110204);