#127922

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#106
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,30,33,43)(20,26)(21,38)(22,39)(23,40)(24,29)(25,32,42,34)(27,37)(28,36)(31,45,35,46)(41,47,44,48);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,22,34,23,31,36,48,24,33,21,32,20,35,37,47)(18,25,28,43,26,44,39,45,29,42,27,30,40,41,38,46);(3,10)(6,9)(11,13)(14,16)(17,20,24,23)(18,26,29,40)(19,31,33,35)(21,36,22,37)(25,41,42,44)(27,38,28,39)(30,45,43,46)(32,47,34,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285490, #72285579, #72285630
maximal normal subgroup of	#72285490, #72285579, #72285630

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35118);