#220189

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127385);