#212333

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 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)(3,8)(4,6)(5,7)(9,10,13,15,12,11,14,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,3,5,4)(2,6,7,8)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,5,3)(2,6,7,8)(9,11,13,16,12,10,14,15)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=egh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,d^c=di,f^c=fh

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, 119529);