#127828

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=gi,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35024);