#127196

order	512
nilpotency class	4
isoclinism	isoclinic class of #7213
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240712, #72240805, #72240806, …
maximal normal subgroup of	#72240712, #72240805, #72240806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34392);