#493303

order	512
nilpotency class	2
isoclinism	isoclinic class of #180872
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

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

544 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)(9,10,14,11,15,20,30,24,16,21,31,19,13,18,29,23)(12,17,27,22,28,35,40,37,25,32,38,34,26,33,39,36);(2,5)(3,7)(9,11,14,18,15,24,30,10,16,19,31,20,13,23,29,21)(12,22,27,33,28,37,40,17,25,34,38,35,26,36,39,32);(1,3,4,7)(2,6,5,8)(9,12,13,26,16,25,15,28)(10,17,18,33,21,32,20,35)(11,22,23,36,19,34,24,37)(14,27,29,39,31,38,30,40);(2,5)(6,8)(12,25)(17,32)(22,34)(26,28)(27,38)(33,35)(36,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313379, #72313392, #72313439, …
maximal normal subgroup of	#72313379, #72313392, #72313439, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400499);