#1634893

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1696 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=hi,d^2=g,e^2=gi,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, 1542089);