#6459837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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:

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

Polycyclic presentation:

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

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, 6367033);