#359166

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1368 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)(5,6,11,15,12,16,10,20)(7,8,18,13,19,23,27,22)(9,14,25,29,26,30,24,33)(17,21,31,28,32,35,36,34);(5,7)(6,13)(8,20)(9,17)(10,27)(11,18)(12,19)(14,28)(15,23)(16,22)(21,33)(24,36)(25,31)(26,32)(29,35)(30,34);(2,4)(5,8,10,22,12,23,11,13)(6,7,15,18,16,19,20,27)(9,21,24,34,26,35,25,28)(14,17,29,31,30,32,33,36);(5,9,11,25,12,26,10,24)(6,14,15,29,16,30,20,33)(7,17,18,31,19,32,27,36)(8,21,13,28,23,35,22,34)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266362);