#5486611

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2686 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,23,24,20)(25,26,29,30)(27,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23)(25,26,29,30)(27,28,31,32);(2,6)(3,8)(10,15)(11,16)(18,22)(20,23)(26,30)(27,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240130, #72240144, #72240187
maximal normal subgroup of	#72240130, #72240144, #72240187
maximal quotient of	#72251705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5393807);