#359371

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,e^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, 266567);