#360369

order	512
nilpotency class	3
isoclinism	isoclinic class of #360368
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2344
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13874
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338603, #72338606, #72338610
maximal normal subgroup of	#72338603, #72338606, #72338610

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267565);