#212395

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 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)(3,8)(4,6)(5,7)(9,10,14,15,13,16,11,12)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31);(1,3,5,4)(2,6,7,8)(9,11,13,14)(10,15,16,12)(18,22)(23,31)(24,29)(25,32);(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,5,3)(2,6,7,8)(9,12,11,16,13,15,14,10)(19,26)(23,31)(25,32)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=egh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,d^c=dh,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, 119591);