#388978

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,21,26,22,27)(19,24,30,25,29,32,20,31);(1,3,6,2)(4,8,5,7)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,24,26,32)(20,23,30,22)(25,27,31,28);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,20,23,19,21,30,22,29)(18,25,28,32,26,31,27,24);(1,5,6,4)(2,7,3,8)(10,14)(13,16)(18,26)(24,32)(25,31)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eh,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,g^c=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, 296174);