#2496604

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#16249	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2906 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fgh,c^2=fg,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327826, #72327858, #72327886
maximal normal subgroup of	#72327826, #72327858, #72327886
maximal quotient of	#72328436

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2403800);