#7007143

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	#45091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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)(13,14,19,20,18,23,17,24)(15,21,31,33,26,37,27,36)(16,28,32,38,30,22,29,34)(25,39,42,44,41,35,40,43);(5,6)(7,11)(8,10)(9,12)(14,20)(16,29)(17,19)(21,33)(23,24)(25,40)(27,31)(30,32)(34,38)(36,37)(41,42)(43,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,25)(17,27)(18,26)(19,31)(20,33)(22,35)(23,37)(24,36)(28,39)(29,40)(30,41)(32,42)(34,43)(38,44);(7,12)(9,11)(13,16,17,29,18,30,19,32)(14,22,24,38,23,28,20,34)(15,25,27,40,26,41,31,42)(21,35,36,44,37,39,33,43);(6,10)(9,11)(14,23)(15,26)(20,24)(22,28)(25,41)(27,31)(34,38)(40,42)

Polycyclic presentation:

2^9|a^2=fgi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914339);