#7504232

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1843 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:

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

Polycyclic presentation:

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

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, 7411428);