#7199180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 7106376);