#6465773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372969);