#6462685

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#28930	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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:

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

Polycyclic presentation:

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

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, 6369881);