#208056

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fi,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115252);