#127788

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	#103
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

425 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)(3,7)(4,6)(5,8)(9,10,13,18,15,21,32,20)(11,22,25,36,23,37,27,40)(12,28,30,17,31,19,14,33)(16,26,35,24,34,39,38,29);(2,6)(3,4)(7,8)(9,11,14,26,32,25,31,24,15,23,30,39,13,27,12,29)(10,16,19,36,18,38,33,22,21,34,28,40,20,35,17,37);(1,3,5,4)(2,7,8,6)(11,23)(16,34)(22,37)(24,29)(25,27)(26,39)(35,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309618, #72309653, #72309658, …
maximal normal subgroup of	#72309618, #72309653, #72309658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34984);