#356641

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1636 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,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, 263837);