#493330

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
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	C₂³	C₂²×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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)(4,7)(5,8)(9,10,15,20,16,22,34,32,17,21,33,31,14,19,13,12)(11,23,27,39,28,18,25,36,29,38,35,30,26,37,40,24);(1,3,6,2)(4,7,5,8)(9,11)(10,18,21,37)(12,30,32,39)(13,25)(14,26)(15,35)(16,28)(17,29)(19,23,22,38)(20,24,31,36)(27,33)(34,40);(1,4,6,5)(2,7,3,8)(9,12,13,19,14,31,33,21,17,32,34,22,16,20,15,10)(11,24,40,37,26,30,35,38,29,36,25,18,28,39,27,23);(1,5,6,4)(2,7,3,8)(9,13,14,33,17,34,16,15)(10,12,19,31,21,32,22,20)(11,25,26,27,29,40,28,35)(18,30,23,36,37,39,38,24)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313248, #72313289, #72313298, …
maximal normal subgroup of	#72313248, #72313289, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400526);