#212029

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 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)(9,10,13,15,12,11,14,16)(17,18,21,25)(19,22,27,31)(20,23,28,24)(26,29,32,30);(2,5)(6,8)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3,4,7)(2,6,5,8)(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(9,11,13,16,12,10,14,15)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119225);