#3464157

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2589 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,29,32,31);(6,8)(10,11)(14,15)(18,23)(19,26)(25,28)(27,32);(5,6)(7,8)(12,16)(14,15)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,29,28,31);(1,2,3,4)(9,11,13,10)(12,14,16,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,31,24,29);(9,12)(10,14)(11,15)(13,16)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304055, #72304090, #72304103
maximal normal subgroup of	#72304055, #72304090, #72304103

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3371353);