#188876

order	512
nilpotency class	3
isoclinism	isoclinic class of #188814
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

943 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^c=gi

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, 96072);