#484133

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33737	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=di,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321227, #72321746, #72321811
maximal normal subgroup of	#72321227, #72321746, #72321811
maximal quotient of	#72323125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391329);