#4598767

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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

1852 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)(5,6,8,10)(7,9,11,12)(13,14,18,16)(15,21,23,19)(17,24,26,20)(22,27,25,28)(29,30,34,37)(31,35,39,32)(33,36,42,44)(38,43,41,40);(5,7,8,11)(6,9,10,12)(14,16)(19,21)(20,24)(27,28)(29,31,34,39)(30,32,37,35)(33,38,42,41)(36,40,44,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,40)(38,42)(43,44);(1,2)(3,4)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,32,34,35)(30,31,37,39)(33,40,42,43)(36,38,44,41);(5,6)(7,9)(8,10)(11,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,33,34,42)(30,36,37,44)(31,38,39,41)(32,40,35,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4505963);