#6628978

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628872
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4540 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^b=ei,g^c=gi,h^c=hi,g^d=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, 6536174);