#5035166

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
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

1930 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=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, 4942362);