#7144028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,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, 7051224);