#245772

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	#45387	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1306 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)(9,10,15,20)(11,21,24,32)(12,25,28,33)(13,19,14,17)(16,29,31,23)(18,27,34,26)(22,30,37,40)(35,38,39,36);(2,5)(3,8)(6,7)(9,11,15,24)(10,16,20,31)(12,22,28,37)(13,29,14,23)(17,32,19,21)(18,30,34,40)(25,35,33,39)(26,38,27,36);(10,17)(13,14)(16,21)(18,33)(19,20)(23,29)(25,34)(26,27)(30,35)(31,32)(36,38)(39,40);(2,5)(3,8)(6,7)(9,12,15,28)(10,18,20,34)(11,22,24,37)(13,27,14,26)(16,30,31,40)(17,33,19,25)(21,35,32,39)(23,38,29,36)

Polycyclic presentation:

2^9|a^2=h,b^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, #72257406, …
maximal normal subgroup of	#72257338, #72257350, #72257406, …
maximal quotient of	#72257933, #72264054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152968);