#482562

order	512
nilpotency class	4
isoclinism	isoclinic class of #482367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1125 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)(3,13,14,23)(4,18,20,7)(5,8)(6,12)(9,17,26,28)(10,19,22,29)(11,15)(16,25)(21,31)(24,30)(27,32);(2,7)(3,14)(4,19)(8,10)(9,24)(11,22)(12,18)(13,25)(16,30)(17,28)(20,29)(21,26)(23,32)(27,31);(2,8)(7,10)(9,23)(11,12)(13,26)(18,22)(21,25)(24,32);(1,3,15,28)(2,9,11,13)(4,16,29,30)(5,17,6,14)(7,21,22,32)(8,23,12,26)(10,25,18,24)(19,31,20,27)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,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, #72238398, #72238401, …
maximal normal subgroup of	#72237055, #72238398, #72238401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389758);