#4414128

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=f,e^2=f,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, 4321324);