#1538070

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

2847 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)(3,7)(4,5)(6,8)(10,13)(14,15)(17,18)(19,23)(20,21)(22,24)(26,29)(27,31);(2,5)(3,8)(9,10)(11,15)(12,13)(14,16)(18,21)(19,24)(25,26,28,29)(27,32,31,30);(2,5)(6,7)(10,13)(14,15)(17,18,20,21)(19,22,24,23)(25,26,28,29)(27,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,4,8)(2,6,5,7)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=fgi,c^2=hi,e^2=fhi,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	#72300706, #72300749, #72300784
maximal normal subgroup of	#72300706, #72300749, #72300784

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1445266);