#220569

order	512
nilpotency class	3
isoclinism	isoclinic class of #220564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61056	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127765);