#6457669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,d^2=f,e^2=i,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,d^c=di

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, 6364865);