#4420147

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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)(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30,33,36)(31,34,37,41)(32,39,40,35)(38,44,43,42);(5,7,9,6)(8,11,12,10)(14,15)(19,21)(20,22)(27,28)(31,37)(34,41)(38,43)(42,44);(5,8)(6,10)(7,11)(9,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,34)(32,38)(33,37)(35,42)(36,41)(39,44)(40,43);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32,33,40)(30,35,36,39)(31,38,37,43)(34,42,41,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,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, 4327343);