#1907213

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2698 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)(3,5)(4,6)(7,8)(9,10)(11,12)(13,15)(14,16)(18,19)(20,24)(25,26,29,32)(27,28,31,30);(1,2,4,6)(3,5,7,8)(9,11,13,16)(10,14,15,12)(18,19)(22,23)(25,27,29,31)(26,30,32,28);(3,7)(5,8)(10,15)(12,14)(17,18,21,19)(20,22,24,23)(25,28,29,30)(26,31,32,27);(1,3,4,7)(2,5,6,8)(9,12,13,14)(10,11,15,16)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326558, #72326591, #72326609
maximal normal subgroup of	#72326558, #72326591, #72326609
maximal quotient of	#72328407

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1814409);