#7197536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=fg,e^2=h,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=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7104732);