#398771

order	512
nilpotency class	3
isoclinism	isoclinic class of #398771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14768
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237785, #72237789, #72237790, …
maximal normal subgroup of	#72237785, #72237789, #72237790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305967);