#2241005

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2832 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,20,21)(19,22,23,24)(25,26,29,32)(27,31,28,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,21)(22,24)(25,26,29,32)(27,30,28,31);(3,7)(6,8)(11,12)(14,16)(17,19,20,23)(18,22,21,24)(25,27,29,28)(26,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=gh,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	#72263168, #72263216, #72263223, …
maximal normal subgroup of	#72263168, #72263216, #72263223, …
maximal quotient of	#72318746, #72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2148201);