#493643

order	512
nilpotency class	4
isoclinism	isoclinic class of #493635
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20609
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#63021	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303256
maximal normal subgroup of	#72303256
maximal quotient of	#72302977, #72310752

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400839);