#7184415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,28,23,27,24,19)(20,25,32,31,30,21,26,29);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19)(18,24)(20,31)(21,26)(22,27)(23,28)(25,32)(29,30);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,29)(21,27)(22,32)(23,30)(24,26)(28,31);(10,13)(14,16)(17,21,24,31,23,25,22,29)(18,26,19,30,27,32,28,20);(9,11,12,15)(10,14,13,16)(18,27)(19,28)(20,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7091611);