#7062693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1860 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)(13,14,18,22)(15,20,24,36)(16,21,29,37)(17,19,30,23)(25,35,28,38)(26,32,40,34)(27,39,31,33)(41,43,42,44);(5,6)(7,11)(8,10)(9,12)(14,19)(15,24)(16,27)(17,30)(20,34)(21,37)(22,23)(25,41)(28,42)(29,31)(32,36)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,26)(18,24)(19,32)(21,38)(22,36)(23,34)(25,29)(27,41)(30,40)(31,42)(33,44)(35,37)(39,43);(7,12)(9,11)(13,16,18,29)(14,21,22,37)(15,25,24,28)(17,31,30,27)(19,33,23,39)(20,35,36,38)(26,41,40,42)(32,43,34,44);(6,10)(9,11)(14,22)(15,24)(19,23)(21,37)(25,28)(26,40)(33,39)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969889);