#6607004

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	C₂×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2309 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=g,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, 6514200);