#7151109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(7,11)(10,12)(13,14,20,17,19,24,18,23)(15,25,34,21,28,38,27,37)(16,29,35,30,31,22,32,33)(26,40,44,41,42,39,43,36);(6,9)(7,11)(13,15,19,28)(14,21,24,37)(16,26,31,42)(17,25,23,38)(18,34,20,27)(22,36,29,41)(30,40,33,39)(32,44,35,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,26)(17,33)(18,32)(19,31)(20,35)(21,36)(23,30)(24,29)(25,39)(27,43)(28,42)(34,44)(37,41)(38,40);(6,9)(10,12)(13,17,20,24,19,23,18,14)(15,21,34,38,28,37,27,25)(16,30,35,22,31,33,32,29)(26,41,44,39,42,36,43,40);(5,7,8,11)(6,10,9,12)(16,31)(22,29)(26,42)(30,33)(32,35)(36,41)(39,40)(43,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7058305);