#6468108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375304);