#7104406

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	#61260	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

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

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, 7011602);