#494805

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402001);