#6403315

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,e^2=hi,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, 6310511);