#6441021

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2334 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,24,28,32)(20,29,31,25)(21,26,27,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,28)(18,24,23,32)(20,30,31,26)(21,25,27,29);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20)(18,25)(19,27)(21,28)(22,31)(23,29)(24,26)(30,32);(17,21,22,27)(18,26,23,30)(19,20,28,31)(24,29,32,25);(19,28)(20,31)(24,32)(25,29)

Polycyclic presentation:

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

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, 6348217);