#4602921

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1667 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)(13,14,18,19)(15,20,22,16)(17,21,26,27)(23,28,25,24)(29,30)(31,35)(32,38)(33,36)(34,37)(39,42)(40,44)(41,43);(6,9)(10,12)(14,19)(15,22)(21,27)(23,25)(29,31,34,32)(30,35,37,38)(33,39,41,40)(36,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,26)(27,28)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,43,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,42)(41,43);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,21,19,27)(15,23,22,25)(16,24,20,28)(29,33,34,41)(30,36,37,43)(31,39,32,40)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=gh,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, 4510117);