#6425622

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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	#52139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1567 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,d^b=dfi,e^b=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, 6332818);