#6438007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
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	C₂⁴	C₂⁵
prime radical	#62859	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4618 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:

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

Polycyclic presentation:

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

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, 6345203);