#212626

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
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	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 119822);