#224648

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eh,d^2=ei,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131844);