#245796

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45267	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257338, #72257350, #72257412, …
maximal normal subgroup of	#72257338, #72257350, #72257412, …
maximal quotient of	#72257935, #72264060

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152992);