#7199773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(3,6)(4,8)(5,7)(9,10,15,16,14,11,13,12)(17,18)(19,23)(20,24)(21,29)(22,25)(26,28)(27,31)(30,32);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,15,16,13)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,26)(29,32);(1,4,5,3)(2,6,7,8)(10,11)(12,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,31)(30,32);(9,12,13,11,14,16,15,10)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(10,11)(12,16)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=gi,d^2=fhi,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^d=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, 7106969);