#6816706

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61294	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723902);