#206012

order	512
nilpotency class	3
isoclinism	isoclinic class of #205799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(9,10,14,18,13,16,12,17)(11,15,21,24,20,22,19,23);(1,2,5,7,6,8,4,3)(9,11,14,19,13,20,12,21)(10,15,17,24,16,22,18,23);(1,3,4,8,6,7,5,2)(10,16)(15,22)(17,18)(23,24);(1,4,6,5)(2,7,8,3)(9,12,13,14)(10,17,16,18)(11,19,20,21)(15,23,22,24)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113208);