#6457916

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365112);