#4970148

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15278	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300713, #72300741, #72300761
maximal normal subgroup of	#72300713, #72300741, #72300761

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4877344);