#358283

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,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, 265479);