#161375

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

848 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239262, #72239295, #72239386
maximal normal subgroup of	#72239262, #72239295, #72239386

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68571);