#3943054

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2876 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=f,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	#72329173, #72329190, #72329211
maximal normal subgroup of	#72329173, #72329190, #72329211
maximal quotient of	#72331283

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3850250);