#7020368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6927564);