#374647

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1313 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281843);