#7172315

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	C₂×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1365 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=f,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7079511);