#208051

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=dh,e^b=ei,g^b=gi

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, 115247);