#493487

order	512
nilpotency class	4
isoclinism	isoclinic class of #493480
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62833	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3486 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2177 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261107, #72261192
maximal normal subgroup of	#72261107, #72261192

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400683);