#493200

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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)(9,10,15,22,16,24,36,20,17,23,35,39,14,21,13,34)(11,25,28,37,29,18,26,31,30,12,19,32,27,33,38,40);(2,5)(6,8)(9,11)(10,18,23,33)(12,21,25,24)(13,26)(14,27)(15,19)(16,29)(17,30)(20,37,34,32)(22,40,39,31)(28,35)(36,38);(9,12,15,32,16,33,36,40,17,25,35,37,14,18,13,31)(10,19,22,27,24,38,20,11,23,28,39,29,21,26,34,30);(1,3,4,7)(2,6,5,8)(9,13,14,35,17,36,16,15)(10,20,21,22,23,34,24,39)(11,26,27,28,30,38,29,19)(12,31,18,37,25,40,33,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313605, #72313642, #72313670, …
maximal normal subgroup of	#72313605, #72313642, #72313670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400396);