#212682

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 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)(10,15)(11,16)(12,13)(17,18,22,26,23,28,21,27)(19,29,20,25,31,32,30,24);(2,5)(6,8)(9,10,12,15,14,16,13,11)(17,19,21,30,23,31,22,20)(18,24,26,29,28,25,27,32);(1,3,4,7)(2,6,5,8)(9,11,12,10,14,15,13,16)(17,19,21,30,23,31,22,20)(18,25,26,32,28,24,27,29);(10,16)(11,15)(17,20,22,31,23,30,21,19)(18,24,26,29,28,25,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119878);