#6432206

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339402);