#7006877

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

2102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=fi,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, 6914073);