#6469698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	#24152	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,26,27,23)(20,22,29,24)(28,31,32,30);(9,10)(11,15)(12,13)(14,16)(18,22)(20,29)(23,30)(24,25)(26,31)(28,32);(2,6)(7,8)(10,13)(14,15)(17,19,21,27)(18,23,25,26)(20,28,29,32)(22,30,24,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(19,27)(23,26)(28,32)(30,31);(1,4,5,3)(2,8,6,7)(19,27)(23,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=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, 6376894);