#494396

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1867
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317388, #72317420, #72317465
maximal normal subgroup of	#72317388, #72317420, #72317465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401592);