#7117805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7025001);