#6425834

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	#2410	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=df,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, 6333030);