#1515686

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1952 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:

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

Polycyclic presentation:

2^9|b^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1422882);