#6606797

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	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 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)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,37)(26,34)(28,38)(29,35)(30,36)(31,42)(33,41)(39,44)(40,43);(5,7)(6,9)(8,12)(10,11)(13,15,17,26,19,28,18,27)(14,20,32,38,23,34,22,24)(16,25,30,40,29,39,31,41)(21,33,42,44,35,43,36,37);(1,2,3,4)(7,12)(9,10)(13,16,19,29)(14,21,23,35)(15,25,28,39)(17,30,18,31)(20,33,34,43)(22,36,32,42)(24,37,38,44)(26,40,27,41);(5,7)(6,10)(8,12)(9,11)(16,29)(21,35)(25,39)(30,31)(33,43)(36,42)(37,44)(40,41);(7,12)(9,10)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,30,29,31)(20,24,34,38)(21,36,35,42)(25,40,39,41)(33,37,43,44)

Polycyclic presentation:

2^9|b^2=fhi,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6513993);