#6372712

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,g^b=gi,f^c=fi,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, 6279908);