#481998

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
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×C₂²⋊C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1122 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)(3,9)(4,17,19,7)(5,8)(6,12)(10,18,21,28)(11,14)(13,25)(15,29,26,20)(16,22)(23,27)(24,30,31,32);(2,7)(4,18)(8,10)(9,20)(11,21)(12,17)(15,30)(19,28)(22,24)(23,31)(25,29)(26,32);(2,8)(3,13)(7,10)(9,23)(11,12)(15,26)(16,27)(17,21)(20,31)(22,25)(24,29)(30,32);(1,3,14,27)(2,9,11,23)(4,15,28,32)(5,16,6,13)(7,20,21,31)(8,22,12,25)(10,24,17,29)(18,30,19,26)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318754
maximal normal subgroup of	#72318754

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389194);