#7232323

order	512
nilpotency class	3
isoclinism	isoclinic class of #7199982
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1244 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32)(33,34,36,40)(35,42,43,38)(37,39,45,41)(44,47,48,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,26)(18,22,23,25)(20,29,30,27)(24,32,28,31)(33,35)(34,38)(36,43)(37,44)(39,46)(40,42)(41,47)(45,48);(19,26)(22,25)(27,29)(31,32)(34,39)(37,45)(38,46)(40,41)(42,47)(44,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(22,25)(24,28)(31,32)(35,43)(38,42)(44,48)(46,47);(2,8)(6,9)(7,16)(14,15)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32)(34,40)(38,42)(39,41)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=fhi,e^2=h,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,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7139519);