#1443614

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3119 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)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,29)(28,31)(30,32);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,14,15,12)(17,18)(19,23)(20,22)(21,24)(25,26,28,31)(27,30,32,29);(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,32)(30,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,11,15,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327607, #72327644, #72327708, …
maximal normal subgroup of	#72327607, #72327644, #72327708, …
maximal quotient of	#72328492, #72330776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1350810);