#6346284

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#3480	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1495 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ci,e^a=eh,f^a=fh,f^b=fh,d^c=dh,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253480);