#6452074

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3257 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:

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

Polycyclic presentation:

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

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, 6359270);