#7197639

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,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, 7104835);