#482588

order	512
nilpotency class	4
isoclinism	isoclinic class of #482367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278825, #72278872, #72278933
maximal normal subgroup of	#72278825, #72278872, #72278933

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389784);