#606728

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1970 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)(7,11)(10,12)(14,18)(15,21)(16,23)(26,28)(30,34)(35,37)(36,40)(42,43);(6,9)(10,12)(14,18)(15,21)(20,25)(22,27)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 513924);