#6465000

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(17,18,22,26,23,21,24,27)(19,28,31,20,32,29,30,25);(2,5)(6,8)(17,19,23,32)(18,25,21,20)(22,30,24,31)(26,29,27,28);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,29,23,25,24,28)(18,19,26,31,21,32,27,30);(17,21,24,26,23,18,22,27)(19,29,30,20,32,28,31,25);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,26,21,27)(19,30,32,31)(20,28,25,29)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6372196);