#7500321

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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)(35,40)(36,42)(37,43)(38,47)(39,44)(41,45)(46,48);(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)(34,40)(36,38)(37,46)(39,45)(41,44)(42,47)(43,48);(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)(33,36,39,46)(34,41,44,40)(35,37,45,38)(42,43,48,47);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,26)(23,25)(27,32)(30,31)(33,36)(34,42)(35,37)(38,45)(39,46)(40,43)(41,47)(44,48);(2,10)(7,11)(8,9)(15,16)(17,20,21,28)(18,24,22,29)(19,27,26,32)(23,30,25,31)(33,37,39,38)(34,43,44,47)(35,36,45,46)(40,42,41,48)

Polycyclic presentation:

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

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, 7407517);