#6771028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6678224);