#7150796

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#44882	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7057992);