#6489319

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,25)(19,27,29,42)(20,24,33,40)(21,35,28,26)(23,36,37,31)(30,43,32,46)(34,48,44,41)(38,47,39,45);(11,12)(14,15)(17,19,28,36,22,29,21,31)(18,23,35,42,25,37,26,27)(20,32,44,45,33,30,34,47)(24,39,48,43,40,38,41,46);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(19,29)(20,33)(23,37)(24,40)(27,42)(31,36)(34,44)(41,48);(9,12,13,11)(10,15,16,14)(17,20,22,33)(18,24,25,40)(19,30,29,32)(21,34,28,44)(23,38,37,39)(26,41,35,48)(27,43,42,46)(31,45,36,47);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(18,25)(20,33)(23,37)(26,35)(27,42)(30,32)(34,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396515);