#6542912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#29600	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450108);