#7604309

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#24129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=hi,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7511505);