#4342131

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1729 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)(14,18)(15,21)(16,23)(26,28)(29,30,34,35)(31,38,39,36)(32,33,43,37)(40,41,42,44);(5,6,9,8)(7,10,11,12)(14,18)(19,24)(20,25)(26,28)(30,35)(31,39)(33,37)(40,42);(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,39)(30,36,35,38)(32,42,43,40)(33,41,37,44);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32)(30,37)(31,40)(33,35)(34,43)(36,41)(38,44)(39,42);(5,8,9,6)(7,12,11,10)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,37)(30,32,35,43)(31,41,39,44)(36,42,38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4249327);