#7500271

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300948
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1296 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407467);