#6519940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
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	#37639	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 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)(13,14,18,27,17,22,16,21)(15,23,24,19,25,20,26,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,22,28)(16,24,18,26)(20,27,23,21);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,26)(17,25)(18,24)(19,21)(22,23)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,27)(15,24,25,26)(19,20,28,23);(6,9)(10,12)(15,25)(19,28)(20,23)(24,26)

Polycyclic presentation:

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

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, 6427136);