#161871

order	512
nilpotency class	3
isoclinism	isoclinic class of #161854
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

775 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dg

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69067);