#2355481

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3066 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)(3,5)(4,8)(6,7)(9,10,14,16)(11,13,15,12)(17,18,20,22)(19,21,23,24)(26,29)(30,32);(2,7)(3,4)(9,11,14,15)(10,12,16,13)(19,23)(21,24)(25,26,28,29)(27,30,31,32);(1,3,6,4)(2,5,7,8)(9,12,14,13)(10,15,16,11)(17,18)(19,21)(20,22)(23,24)(25,27,28,31)(26,30,29,32);(1,4,6,3)(2,5,7,8)(9,13,14,12)(10,15,16,11)(17,19,20,23)(18,21,22,24)(25,26)(27,30)(28,29)(31,32);(1,5)(2,3)(4,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=fghi,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, #72263212, #72263215, …
maximal normal subgroup of	#72263168, #72263212, #72263215, …
maximal quotient of	#72318552, #72329202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2262677);