#3192277

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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2845 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=fhi,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	#72328102, #72328155, #72328172
maximal normal subgroup of	#72328102, #72328155, #72328172
maximal quotient of	#72328751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3099473);