#4385522

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1856 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)(9,12)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26)(29,30)(31,32)(33,37)(34,35)(36,39)(38,40)(41,44)(42,43);(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,23)(22,27)(30,35)(32,36)(37,43)(40,44);(1,2)(3,4)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,31)(30,36)(32,35)(33,41)(34,39)(37,40)(38,42)(43,44);(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28)(29,32,34,36)(30,31,35,39)(33,40,42,44)(37,38,43,41);(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)(29,33)(30,37)(31,38)(32,40)(34,42)(35,43)(36,44)(39,41)

Polycyclic presentation:

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

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, 4292718);