#4414511

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 8388608
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3830 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)(6,8)(11,15)(14,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31);(10,13)(14,16)(17,19,22,27)(18,23,20,25)(21,26,30,29)(24,31,28,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31);(1,2)(3,4)(5,6)(7,8)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31);(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4321707);