#6444697

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fg,d^2=f,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, 6351893);