#6448607

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	#34935	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3464 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355803);