#512728

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₁₆	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,11,16,12,17,19,28,13,18,20,29,22,30,31,38,14,15,21,25,23,26,32,35,24,27,33,36,34,37,39,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)

Polycyclic presentation:

2^9|a^2=f,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231858, #72231859, #72231862, …
maximal normal subgroup of	#72231858, #72231859, #72231862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419924);