#6411956

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1679 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6319152);