#5880911

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2553 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=fgi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336441, #72336449, #72336475
maximal normal subgroup of	#72336441, #72336449, #72336475

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5788107);