#128270

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216232, #72216448, #72225466
maximal normal subgroup of	#72216232, #72216448, #72225466
maximal quotient of	#72225546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35466);