#1271387

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3833 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(18,21)(19,23)(25,26)(27,30)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,32)(30,31);(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,3)(2,6,7,8)(9,12,13,10)(11,15,16,14)(17,18)(19,24)(20,21)(22,23)(25,26)(27,29)(28,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|b^2=fhi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232588, #72232622, #72232649
maximal normal subgroup of	#72232588, #72232622, #72232649
maximal quotient of	#72328427

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1178583);