#6444782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	#25759	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

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

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, 6351978);