#356642

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

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,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, 263838);