#7160940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45079	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7068136);