#480666

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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,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	#72282993, #72283025, #72283098, …
maximal normal subgroup of	#72282993, #72283025, #72283098, …
maximal quotient of	#72284262

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387862);