#3325189

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
prime radical	#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3022 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)(3,6)(4,5)(7,8)(10,14)(12,16)(17,18)(19,24)(20,22)(21,23)(25,26)(27,30)(28,29)(31,32);(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,21)(20,24)(22,23)(26,29)(27,31);(3,7)(6,8)(10,14)(11,15)(18,20)(19,23)(25,26,28,29)(27,30,31,32);(1,3,5,7)(2,6,4,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,18)(19,21,23,24)(25,26)(27,32)(28,29)(30,31);(1,4,5,2)(3,6,7,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,23)(19,20)(22,24)(25,27,28,31)(26,30,29,32)

Polycyclic presentation:

2^9|c^2=gi,d^2=fhi,e^2=fghi,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	#72334192, #72334200, #72334234
maximal normal subgroup of	#72334192, #72334200, #72334234

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3232385);