#6383687

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#392	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290883);