#492848

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=be,e^a=ei,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226782, #72226794, #72226797
maximal normal subgroup of	#72226782, #72226794, #72226797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400044);