#7273026

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42505	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

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, 7180222);