#857202

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3622 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328664, #72328669, #72328707
maximal normal subgroup of	#72328664, #72328669, #72328707
maximal quotient of	#72331331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 764398);