#480662

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#120	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1393 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283413, #72283435, #72283524, …
maximal normal subgroup of	#72283413, #72283435, #72283524, …
maximal quotient of	#72284262

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387858);