#6545002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 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)(13,14,19,31,18,23,17,22)(15,20,32,42,25,34,26,35)(16,27,28,21,30,36,29,37)(24,38,40,33,39,44,41,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,20,23,34)(16,24,30,39)(17,26,19,32)(21,33,37,43)(22,35,31,42)(27,38,36,44)(28,40,29,41);(6,9)(10,12)(13,16,18,30)(14,21,23,37)(15,24,25,39)(17,28,19,29)(20,33,34,43)(22,36,31,27)(26,40,32,41)(35,44,42,38);(5,7)(6,10)(8,11)(9,12)(15,25)(20,34)(24,39)(26,32)(33,43)(35,42)(38,44)(40,41);(7,11)(10,12)(13,17,18,19)(14,22,23,31)(15,26,25,32)(16,28,30,29)(20,35,34,42)(21,36,37,27)(24,40,39,41)(33,44,43,38)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452198);