#6516827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×#3554	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424023);