#6345862

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61434	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,e^d=ehi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321691, #72322026, #72322038
maximal normal subgroup of	#72321691, #72322026, #72322038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253058);