#6483275

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 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)(7,12)(8,10)(9,11)(13,14,19,24)(15,25,29,40)(16,21,17,22)(18,32,34,23)(20,33,37,28)(26,41,27,42)(30,39,31,38)(35,44,36,43);(7,8)(10,12)(13,15,19,29)(14,20,24,37)(16,26,17,27)(18,33,34,28)(21,35,22,36)(23,40,32,25)(30,44,31,43)(38,42,39,41);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,19)(18,30)(20,35)(22,24)(23,38)(25,41)(27,29)(28,43)(31,34)(32,39)(33,44)(36,37)(40,42);(5,8,9,7)(6,10,11,12)(13,17,19,16)(14,22,24,21)(15,27,29,26)(18,31,34,30)(20,36,37,35)(23,39,32,38)(25,42,40,41)(28,44,33,43);(6,11)(10,12)(16,17)(21,22)(26,27)(30,31)(35,36)(38,39)(41,42)(43,44)

Polycyclic presentation:

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

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, 6390471);