#482214

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264805, #72264821, #72264826, …
maximal normal subgroup of	#72264805, #72264821, #72264826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389410);