#356643

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

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(9,10,14,18,15,19,13,28)(11,20,22,16,23,31,29,32)(12,17,27,35,24,33,26,40)(21,36,37,30,25,38,39,34);(1,2)(3,4)(5,6,7,8)(12,24)(17,33)(21,25)(26,27)(30,34)(35,40)(36,38)(37,39);(9,11,14,22,15,23,13,29)(10,16,18,31,19,32,28,20)(12,25,27,39,24,21,26,37)(17,34,35,36,33,30,40,38);(2,4)(9,12,14,27,15,24,13,26)(10,17,18,35,19,33,28,40)(11,21,22,37,23,25,29,39)(16,30,31,38,32,34,20,36)

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=dei,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, 263839);