#497383

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3280	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1184 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:

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

Polycyclic presentation:

2^9|a^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,c^b=ci,d^b=de,f^c=fh,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404579);