#228168

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#18520	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

684 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=de,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135364);