#482586

order	512
nilpotency class	4
isoclinism	isoclinic class of #482367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂×#392	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

2^9|a^2=f,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	#72237055, #72238399, #72238400, …
maximal normal subgroup of	#72237055, #72238399, #72238400, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389782);