#231497

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138693);