#6459774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3138 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,28)(17,22)(18,33)(19,24)(20,26)(23,29)(25,35)(27,36)(30,41)(31,37)(32,43)(34,40)(38,42)(39,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,27,29,16)(14,20,23,21,24,28,33,36)(17,25,32,40,31,39,42,30)(22,34,38,35,37,41,43,44);(6,10)(7,11)(13,16,29,27,19,26,18,15)(14,21,33,20,24,36,23,28)(17,30,42,39,31,40,32,25)(22,35,43,34,37,44,38,41);(7,11)(9,12)(13,17)(14,22)(15,25)(16,30)(18,32)(19,31)(20,34)(21,35)(23,38)(24,37)(26,40)(27,39)(28,41)(29,42)(33,43)(36,44);(17,31)(22,37)(25,39)(30,40)(32,42)(34,41)(35,44)(38,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366970);