#6425704

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	#52160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1584 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 6332900);