#493216

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216223, #72216232, #72216233, …
maximal normal subgroup of	#72216223, #72216232, #72216233, …
maximal quotient of	#72225524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400412);