#7193172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996395
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	#24174	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 7100368);