#3228915

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2951 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326558, #72326577, #72326638
maximal normal subgroup of	#72326558, #72326577, #72326638

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3136111);