#4420532

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,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, 4327728);