#4341796

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3342 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)(6,7)(11,15)(14,16)(17,18,22,25)(19,26,29,23)(20,30,24,21)(27,32,31,28);(10,13)(14,16)(17,19,22,29)(18,23,25,26)(20,27,24,31)(21,32,30,28);(5,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,27)(25,26)(30,32);(1,2,3,4)(5,7,8,6)(17,20,22,24)(18,21,25,30)(19,27,29,31)(23,32,26,28);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,28)(20,25)(22,30)(23,27)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4248992);