#6403325

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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

1749 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)(16,27,30,43)(18,32,29,23)(20,35,37,31)(24,39,28,42)(26,34,40,38)(33,41,36,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,19,22,21)(16,28,30,24)(18,26,29,40)(20,36,37,33)(23,34,32,38)(27,39,43,42)(31,44,35,41);(13,16,29,35,17,30,18,31)(14,20,32,43,22,37,23,27)(15,24,40,44,25,28,26,41)(19,33,38,39,21,36,34,42);(7,12)(9,11)(13,15)(14,21)(16,28)(17,25)(18,26)(19,22)(20,33)(23,38)(24,30)(27,42)(29,40)(31,44)(32,34)(35,41)(36,37)(39,43);(1,3,4,2)(6,10)(9,11)(14,22)(15,25)(16,30)(23,32)(26,40)(31,35)(33,36)(39,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,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^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310521);