#386242

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293438);