#6469899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(3,7)(4,8)(5,6)(17,18,22,27,23,28,20,25)(19,29,30,39,32,42,37,24)(21,34,35,40,36,26,33,41)(31,45,44,38,47,43,46,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,28,39)(20,30,22,37)(21,31,36,47)(25,29,27,42)(26,38,34,48)(33,44,35,46)(40,43,41,45);(2,6)(7,8)(10,13)(14,16)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,35,36,33)(24,29,39,42)(26,41,34,40)(31,46,47,44)(38,45,48,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,21,23,36)(18,26,28,34)(19,31,32,47)(20,33,22,35)(24,38,39,48)(25,40,27,41)(29,43,42,45)(30,44,37,46);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(21,36)(26,34)(31,47)(33,35)(38,48)(40,41)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377095);