#4420568

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

3606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,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, 4327764);