#493199

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
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×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

611 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)(9,10,15,22,16,23,34,32,17,12,19,33,14,21,13,20)(11,24,28,40,29,18,26,38,30,25,35,37,27,31,36,39);(2,5)(6,8)(9,11)(10,18,12,31)(13,26)(14,27)(15,35)(16,29)(17,30)(19,28)(20,37,32,40)(21,24,23,25)(22,39,33,38)(34,36);(9,12,15,33,16,21,34,20,17,10,19,22,14,23,13,32)(11,25,28,37,29,31,26,39,30,24,35,40,27,18,36,38);(1,3,4,7)(2,6,5,8)(9,13,14,19,17,34,16,15)(10,20,21,33,12,32,23,22)(11,26,27,28,30,36,29,35)(18,37,24,38,31,40,25,39)

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,g^a=gi,c^b=ch,f^b=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313255, #72313273, #72313298, …
maximal normal subgroup of	#72313255, #72313273, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400395);