#6458756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=i,d^2=g,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, 6365952);