#6811676

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37682	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2099 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:

(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,30)(24,33)(25,32)(26,34)(27,36)(28,38)(29,35)(31,39)(37,42)(40,44)(41,43);(2,3)(5,7,10,6)(8,12,9,11)(13,15,18,24)(14,19,22,33)(16,27,28,25)(17,26,30,39)(20,34,23,31)(21,36,38,32)(29,41,42,40)(35,43,37,44);(1,2,4,3)(14,20)(15,24)(17,30)(19,31)(21,35)(22,23)(25,27)(29,42)(32,43)(33,34)(36,44)(37,38);(5,8,10,9)(6,11,7,12)(13,16)(14,21)(15,25)(17,29)(18,28)(19,32)(20,35)(22,38)(23,37)(24,27)(26,40)(30,42)(31,43)(33,36)(34,44)(39,41);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,22)(15,24)(16,28)(20,23)(26,39)(29,42)(32,36)(43,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=hi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

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, 6718872);