#500048

order	512
nilpotency class	4
isoclinism	isoclinic class of #499855
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#32890	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340704, #72340712, #72340715
maximal normal subgroup of	#72340704, #72340712, #72340715

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407244);