#3968157

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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

2778 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)(3,7)(4,5)(6,8)(10,12)(11,15)(18,22)(20,23)(25,26)(27,32)(28,31)(29,30);(2,5)(3,8)(9,10)(11,16)(12,13)(14,15)(17,18,21,22)(19,20,24,23)(26,30)(31,32);(3,8)(6,7)(10,12)(14,16)(18,22)(20,23)(25,27,29,28)(26,31,30,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,24)(18,23,22,20)(25,28,29,27)(26,32,30,31);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,15,16)(17,20,21,23)(18,19,22,24)(25,26)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=hi,e^2=fg,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	#72328607, #72328620, #72328721
maximal normal subgroup of	#72328607, #72328620, #72328721
maximal quotient of	#72330946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3875353);