#6520354

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(13,14,19,24)(15,25,29,35)(16,30,31,38)(17,23,18,22)(20,28,36,27)(21,32,37,33)(26,34,40,44)(39,41,43,42);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,28,19,29,17,27)(14,20,23,35,24,36,22,25)(16,26,33,42,31,40,32,41)(21,34,30,43,37,44,38,39);(6,9)(7,11)(13,16,18,33,19,31,17,32)(14,21,23,30,24,37,22,38)(15,26,28,42,29,40,27,41)(20,34,35,43,36,44,25,39);(5,7,8,11)(6,10,9,12)(16,31)(21,37)(26,40)(30,38)(32,33)(34,44)(39,43)(41,42);(6,9)(10,12)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427550);