#6816705

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

Polycyclic presentation:

2^9|c^2=gh,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, 6723901);