#7236754

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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

1497 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:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7143950);