#7124126

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	#3493	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)(17,18,23,27)(19,30,34,45)(20,26,36,44)(21,29,22,28)(24,37,41,33)(25,38,42,35)(31,39,48,40)(32,46,47,43);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,23,34)(18,24,27,41)(20,31,36,48)(21,37,22,33)(25,39,42,40)(26,43,44,46)(28,45,29,30)(32,38,47,35);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,31)(21,35)(22,38)(23,36)(24,39)(26,28)(27,42)(29,44)(30,46)(32,37)(33,47)(34,48)(40,41)(43,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,32)(21,38)(22,35)(23,36)(24,40)(25,28)(27,44)(29,42)(30,46)(31,37)(33,48)(34,47)(39,41)(43,45);(2,8)(6,9)(7,16)(14,15)(18,27)(24,41)(25,42)(26,44)(28,29)(30,45)(39,40)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,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, 7031322);