#202479

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 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:

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

Polycyclic presentation:

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

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, 109675);