#356745

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,d^c=di,f^c=fi

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, 263941);