#6606908

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2266 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)(7,10)(8,11)(9,12)(13,14,19,23)(15,24,27,39)(16,21,30,36)(17,22,18,32)(20,26,34,28)(25,38,41,43)(29,35,31,37)(33,42,44,40);(5,7)(6,9)(8,12)(10,11)(13,15,18,28,19,27,17,26)(14,20,22,24,23,34,32,39)(16,25,29,40,30,41,31,42)(21,33,37,38,36,44,35,43);(1,2,3,4)(7,12)(9,10)(13,16,19,30)(14,21,23,36)(15,25,27,41)(17,31,18,29)(20,33,34,44)(22,37,32,35)(24,38,39,43)(26,42,28,40);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,28)(16,29,30,31)(20,24,34,39)(21,35,36,37)(25,40,41,42)(33,43,44,38);(7,12)(9,10)(15,27)(16,30)(20,34)(21,36)(24,39)(26,28)(29,31)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514104);