#7503156

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2715 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410352);