#3967538

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2864 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)(3,8)(4,7)(5,6)(9,10,12,14)(11,15,16,13)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,31,32,29);(2,6)(7,8)(9,10,12,14)(11,13,16,15)(18,21)(19,24)(27,32)(29,31);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23)(25,26)(27,29)(28,30)(31,32);(1,4,5,3)(2,8,6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,28,32)(26,29,30,31);(1,2)(3,8)(4,7)(5,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,24)(21,22)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=f,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232607, #72232687, #72232692
maximal normal subgroup of	#72232607, #72232687, #72232692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3874734);