#6371428

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(5,6)(7,8)(9,10,14,16,15,20,13,19)(11,21,24,17,25,31,23,32)(12,18,28,29,26,33,27,34)(22,35,38,30,36,40,37,39);(1,2,3,4)(10,16)(13,14)(17,21)(18,29)(19,20)(23,24)(27,28)(30,35)(31,32)(33,34)(37,38)(39,40);(6,8)(9,11,15,25)(10,17,20,32)(12,22,26,36)(13,24,14,23)(16,21,19,31)(18,30,33,39)(27,38,28,37)(29,35,34,40);(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,29)(17,30)(19,34)(20,33)(21,35)(23,37)(24,38)(25,36)(31,40)(32,39);(12,26)(18,33)(22,36)(27,28)(29,34)(30,39)(35,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278624);