#6459797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3063 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,43)(16,30,33,37)(17,22,31,36)(18,35,26,23)(20,32,38,28)(21,39,40,27)(34,42,44,41);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,26,32,19,29,18,28)(14,20,35,43,24,38,23,25)(16,31,39,34,33,17,27,44)(21,22,37,42,40,36,30,41);(6,10)(7,11)(13,16,26,39,19,33,18,27)(14,21,35,37,24,40,23,30)(15,17,32,44,29,31,28,34)(20,36,43,41,38,22,25,42);(7,11)(9,12)(13,17)(14,22)(15,27)(16,32)(18,34)(19,31)(20,37)(21,25)(23,41)(24,36)(26,44)(28,33)(29,39)(30,38)(35,42)(40,43);(16,33)(17,31)(21,40)(22,36)(27,39)(30,37)(34,44)(41,42)

Polycyclic presentation:

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

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, 6366993);