#7182404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#3460	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7089600);