#6345411

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,29)(26,35)(28,41)(30,40)(31,39)(32,45)(34,37)(36,43)(38,44)(42,48)(46,47);(17,19,22,30)(18,23,25,37)(20,28,31,42)(21,34,35,29)(24,36,39,47)(26,40,33,27)(32,46,44,43)(38,48,45,41);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,20,22,31)(18,24,25,39)(19,28,30,42)(21,32,35,44)(23,36,37,47)(26,38,33,45)(27,41,40,48)(29,43,34,46);(3,9)(6,12)(10,15)(13,16)(18,25)(20,31)(23,37)(26,33)(27,40)(28,42)(32,44)(43,46);(2,7)(6,12)(8,14)(13,16)(18,25)(20,31)(23,37)(26,33)(27,40)(28,42)(32,44)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252607);