#358371

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,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, 265567);