#2073545

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2084 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fhi,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304046, #72304071, #72304113
maximal normal subgroup of	#72304046, #72304071, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1980741);