#7144060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44928	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(11,15)(14,16)(18,23)(20,21)(24,30)(25,29)(26,28)(31,32);(1,3,4,8)(2,5,6,7)(10,13)(11,15);(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,24,27,32)(20,23,21,29)(26,31,28,30);(3,8)(5,7)(10,13)(14,16)(17,19)(18,24)(20,28)(21,26)(22,27)(23,30)(25,32)(29,31);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7051256);