#6487617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4032 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)(7,10)(8,11)(9,12)(13,14,19,22)(15,25,27,37)(16,30,34,40)(17,24,18,23)(20,29,36,28)(21,32,39,33)(26,38,31,35)(41,42,44,43);(2,4)(7,8)(10,11)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,36)(23,37)(24,25)(26,34)(30,41)(32,42)(33,43)(35,39)(40,44);(5,7,9,8)(6,10,12,11)(13,16,19,34)(14,21,22,39)(15,26,27,31)(17,33,18,32)(20,35,36,38)(23,30,24,40)(25,41,37,44)(28,43,29,42);(5,8,9,7)(6,11,12,10)(14,22)(20,36)(21,39)(23,24)(25,37)(30,40)(35,38)(41,44);(6,12)(10,11)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394813);