#6483495

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2090 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)(15,27)(16,33)(17,23)(18,24)(19,25)(20,26)(21,30)(22,34)(28,38)(29,42)(31,41)(32,40)(35,39)(36,43)(37,44);(1,2,3,4)(7,8)(10,12)(13,15,19,31,20,32,18,30)(14,21,24,40,26,41,25,27)(16,28,34,37,35,17,29,36)(22,38,33,43,42,23,39,44);(5,7,9,8)(6,10,11,12)(13,16,18,29,20,35,19,34)(14,22,25,39,26,42,24,33)(15,28,30,36,32,17,31,37)(21,38,27,44,41,23,40,43);(5,8,9,7)(6,10,11,12)(13,17,20,28)(14,23,26,38)(15,29,32,34)(16,30,35,31)(18,37,19,36)(21,39,41,33)(22,27,42,40)(24,44,25,43);(6,11)(10,12)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,34,35,29)(17,36,28,37)(21,40,41,27)(22,39,42,33)(23,43,38,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,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, 6390691);