#7006397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2009 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913593);