#128257

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(9,10,14,20,15,21,36,50,16,22,37,51,13,19,34,48)(11,23,26,52,25,41,55,49,27,47,35,63,33,17,38,40)(12,18,30,44,32,46,62,71,31,45,61,70,29,43,59,68)(24,42,53,66,28,56,57,72,54,67,64,69,58,39,60,65);(1,2,3,5,4,6,7,8)(9,11)(10,17)(12,28,31,58)(13,33)(14,35)(15,25)(16,27)(18,42,45,67)(19,47)(20,49)(21,23)(22,41)(24,32,54,29)(26,37)(30,60,61,57)(34,55)(36,38)(39,46,56,43)(40,51)(44,69,70,66)(48,52)(50,63)(53,59,64,62)(65,68,72,71);(9,12,14,30,15,32,36,62,16,31,37,61,13,29,34,59)(10,18,20,44,21,46,50,71,22,45,51,70,19,43,48,68)(11,24,26,53,25,28,55,57,27,54,35,64,33,58,38,60)(17,39,40,65,23,42,52,66,41,56,49,72,47,67,63,69)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35453);