#3078324

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
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

1706 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)(13,14,18,16)(15,21,22,19)(17,20,25,24)(23,27,28,26)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(6,9)(10,12)(14,16)(15,22)(17,25)(26,27)(29,31,34,32)(30,35,37,38)(33,40,42,39)(36,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(13,16,18,14)(15,19,22,21)(17,20,25,24)(23,27,28,26)(29,30)(31,35)(32,38)(33,41)(34,37)(36,42)(39,44)(40,43);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

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, 2985520);