#7182417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2006	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi

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, 7089613);