#6380999

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288195);