#7124134

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

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,d^a=di,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, 7031330);