#127765

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

455 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=i,f^2=ghi,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	#72309655, #72309662, #72309670, …
maximal normal subgroup of	#72309655, #72309662, #72309670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34961);