#6483077

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37301	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(18,19)(21,22)(24,27)(25,26)(28,32)(29,30);(2,5)(6,8)(10,14)(11,12)(17,18,22,25,23,26,21,19)(20,28,29,27,31,24,30,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,19,22,18,23,25,21,26)(20,27,29,24,31,32,30,28);(9,12,13,11)(10,16,14,15)(17,20,23,31)(18,24,26,28)(19,27,25,32)(21,30,22,29);(10,14)(15,16)(17,21,23,22)(18,19,26,25)(20,29,31,30)(24,32,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390273);