#7124130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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	#3492	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,24,29)(19,30,34,42)(20,35,40,44)(21,27,25,33)(22,28,23,32)(26,39,45,38)(31,41,36,37)(43,48,46,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,24,34)(18,25,29,21)(20,36,40,31)(22,27,23,33)(26,41,45,37)(28,42,32,30)(35,48,44,47)(38,43,39,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20,24,40)(18,26,29,45)(19,31,34,36)(21,41,25,37)(22,38,23,39)(27,46,33,43)(28,44,32,35)(30,47,42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,25)(18,27,29,33)(19,32,34,28)(20,37,40,41)(22,42,23,30)(26,43,45,46)(31,44,36,35)(38,47,39,48);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,28,29,32)(19,33,34,27)(20,38,40,39)(21,42,25,30)(26,44,45,35)(31,43,36,46)(37,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7031326);