#6483379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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	#30253	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2092 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)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,30)(27,40)(28,41)(29,37)(31,42)(32,38)(33,39)(35,43)(36,44);(7,8)(10,12)(13,15,19,30,20,31,18,29)(14,21,24,26,25,37,34,42)(16,27,32,43,17,28,33,44)(22,35,39,40,23,36,38,41);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,22)(15,27)(17,20)(18,33)(19,32)(21,35)(23,25)(24,39)(26,40)(28,31)(29,44)(30,43)(34,38)(36,37)(41,42);(5,8,9,7)(6,10,11,12)(13,17,20,16)(14,23,25,22)(15,28,31,27)(18,32,19,33)(21,36,37,35)(24,38,34,39)(26,41,42,40)(29,43,30,44);(6,11)(10,12)(13,18,20,19)(14,24,25,34)(15,29,31,30)(16,32,17,33)(21,26,37,42)(22,38,23,39)(27,43,28,44)(35,41,36,40)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=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, 6390575);