#6629690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2419 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:

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

Polycyclic presentation:

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

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, 6536886);