#162325

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
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 ▷ C₂³ ▷ C₁

843 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cg,d^a=dgh,e^a=eg,c^b=ci,e^b=eh,f^b=fgh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69521);