#358451

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 265647);