#7006871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 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)(7,9)(8,10)(11,12)(13,14,17,23,18,22,29,19)(15,20,26,33,25,34,39,30)(16,21,27,36,28,35,40,31)(24,32,37,42,38,43,44,41);(5,6)(7,11)(8,10)(9,12)(14,19)(16,27)(17,29)(20,30)(22,23)(24,37)(26,39)(28,40)(31,36)(33,34)(38,44)(41,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,30)(21,32)(22,34)(23,33)(27,37)(28,38)(29,39)(31,41)(35,43)(36,42)(40,44);(7,12)(9,11)(13,16,17,27,18,28,29,40)(14,21,23,36,22,35,19,31)(15,24,26,37,25,38,39,44)(20,32,33,42,34,43,30,41);(6,10)(9,11)(14,22)(15,25)(19,23)(21,35)(24,38)(26,39)(31,36)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914067);