#6494417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25777	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2450 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)(7,11)(10,12)(13,14,19,23)(15,25,28,43)(16,29,33,38)(17,22,30,37)(18,36,31,24)(20,32,39,27)(21,34,40,26)(35,42,44,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,28,31,32)(14,20,24,25,23,39,36,43)(16,30,26,44,33,17,34,35)(21,22,29,41,40,37,38,42);(6,9)(10,12)(13,16,31,34,19,33,18,26)(14,21,36,38,23,40,24,29)(15,17,32,44,28,30,27,35)(20,37,43,41,39,22,25,42);(6,9)(10,12)(13,17)(14,22)(15,26)(16,32)(18,35)(19,30)(20,38)(21,25)(23,37)(24,41)(27,33)(28,34)(29,39)(31,44)(36,42)(40,43);(5,7,8,11)(6,10,9,12)(14,23)(16,33)(17,30)(20,39)(24,36)(25,43)(26,34)(35,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6401613);