#217683

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
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	C₂³	C₂×C₄×D₄
prime radical	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 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)(3,7)(4,6)(5,8)(10,13)(11,14)(15,16)(17,18,21,25,22,20,26,24)(19,23,28,31,29,27,32,30);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,14,15,16)(17,19,26,32,22,29,21,28)(18,23,24,30,20,27,25,31);(3,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,27)(21,24)(23,29)(25,26)(28,30)(31,32);(2,6)(5,7)(9,10,12,13)(11,14,15,16)(17,21,22,26)(18,24,20,25)(19,28,29,32)(23,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124879);