#6370846

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278042);