#7124138

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

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,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, 7031334);