#6425818

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	#34829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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)(17,18,22,24,21,25,20,29)(19,23,28,30,27,31,26,32);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32);(1,3,5,4)(2,6,7,8)(10,13)(14,15)(17,20,21,22)(18,24,25,29)(19,26,27,28)(23,30,31,32);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,18,20,29,21,25,22,24)(19,23,26,32,27,31,28,30);(9,11,12,16)(10,14,13,15)(19,27)(23,31)(26,28)(30,32)

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=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di

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, 6333014);