#495770

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	C₄×B(4,𝔽₂)
prime radical	#33567	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=g,d^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,c^b=ci,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	#72231703, C₂×#495770, #72317420
maximal normal subgroup of	#72231703, C₂×#495770, #72317420
maximal quotient of	#72257236, #72314023, C₂×#495770

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402966);