#6536854

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(17,18,22,26)(19,27,30,43)(20,32,34,24)(21,36,28,25)(23,37,39,31)(29,45,33,42)(35,41,44,47)(38,46,40,48);(9,10)(11,14)(12,13)(15,16)(17,19,28,37,22,30,21,31)(18,23,36,43,26,39,25,27)(20,33,44,46,34,29,35,48)(24,40,47,42,32,38,41,45);(2,6)(7,8)(10,13)(14,16)(17,20)(18,24)(19,29)(21,35)(22,34)(23,38)(25,41)(26,32)(27,42)(28,44)(30,33)(31,46)(36,47)(37,48)(39,40)(43,45);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,34)(23,39)(24,32)(27,43)(31,37)(35,44)(41,47);(1,4,5,3)(2,8,6,7)(11,15)(14,16)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444050);