#6386771

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2034 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)(7,11)(8,10)(9,12)(13,14)(15,22)(16,29)(17,25)(18,24)(19,23)(20,28)(21,33)(26,41)(27,36)(30,43)(31,39)(32,38)(34,40)(35,44)(37,42);(5,7)(6,9)(8,12)(10,11)(13,15,18,28)(14,20,24,22)(16,30,32,26)(17,34,19,27)(21,37,39,35)(23,40,25,36)(29,41,38,43)(31,42,33,44);(13,16,19,33,18,32,17,31)(14,21,25,29,24,39,23,38)(15,26,27,42,28,30,34,44)(20,35,36,43,22,37,40,41);(7,12)(9,11)(13,15)(14,22)(16,30)(17,34)(18,28)(19,27)(20,24)(21,35)(23,36)(25,40)(26,32)(29,43)(31,42)(33,44)(37,39)(38,41);(6,10)(9,11)(13,17,18,19)(14,23,24,25)(15,27,28,34)(16,31,32,33)(20,36,22,40)(21,38,39,29)(26,42,30,44)(35,43,37,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293967);