#392378

order	512
nilpotency class	3
isoclinism	isoclinic class of #392203
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2280
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^c=hi,g^d=gi

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, 299574);