#7487840

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42679	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1962 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,23,28,37)(20,31,32,25)(21,24,34,26)(29,43,41,36)(30,38,42,35)(33,40,45,39)(44,46,48,47);(2,5)(3,7)(10,14)(11,12)(17,19,22,28)(18,23,27,37)(20,29,32,41)(21,30,34,42)(24,38,26,35)(25,36,31,43)(33,44,45,48)(39,47,40,46);(9,10)(11,16)(12,15)(13,14)(18,24)(19,28)(20,32)(21,34)(23,35)(25,40)(26,27)(31,39)(36,47)(37,38)(43,46)(44,48);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,39)(26,40)(27,31)(28,41)(30,44)(34,45)(35,46)(37,43)(38,47)(42,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,28)(20,32)(23,37)(25,31)(30,42)(33,45)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7395036);