#6468223

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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

1572 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,26,22,25,35,40)(19,27,29,37,30,41,34,23)(20,31,33,47,32,24,38,39)(28,43,45,36,44,42,48,46);(10,14)(11,12)(17,19,22,30)(18,23,25,37)(20,28,32,44)(21,34,35,29)(24,36,31,46)(26,41,40,27)(33,48,38,45)(39,43,47,42);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,31)(26,39)(27,42)(29,45)(30,44)(34,48)(35,38)(37,46)(40,47)(41,43);(9,12,13,11)(10,16,14,15)(18,25)(23,37)(24,31)(26,40)(27,41)(36,46)(39,47)(42,43);(1,3,4,7)(2,6,5,8)(20,32)(24,31)(28,44)(33,38)(36,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6375419);