#6579322

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

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

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, 6486518);