#5989722

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2690 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)(3,7,8,5)(9,10,14,12)(11,13,15,16)(17,18,21,22)(19,24,20,23)(25,26)(27,30)(28,32)(29,31);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(18,22)(23,24)(25,27,29,32)(26,28,31,30);(10,12)(11,15)(17,19,21,20)(18,23,22,24)(25,27,29,32)(26,30,31,28);(1,3)(2,5)(4,8)(6,7)(9,12,14,10)(11,13,15,16)(17,20,21,19)(18,24,22,23)(25,27)(26,28)(29,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,27)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgh,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334176, #72334188, #72334217
maximal normal subgroup of	#72334176, #72334188, #72334217

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5896918);