#6491265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398461);