#483478

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	C₂²×#107
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260491, #72321227, #72321901
maximal normal subgroup of	#72260491, #72321227, #72321901
maximal quotient of	#72323099

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390674);