#6441761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	#35533	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2328 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)(7,9)(8,10)(11,12)(13,14,17,22,19,15,18,20)(16,21,25,28,26,23,24,27);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,20)(15,22)(16,24)(17,18)(21,23)(25,26);(2,4)(6,10)(9,12)(13,15,17,20,19,14,18,22)(16,23,25,27,26,21,24,28);(2,4)(13,16,17,25,19,26,18,24)(14,21,22,28,15,23,20,27);(14,15)(20,22)(21,23)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348957);