#6370372

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,25,29,26,30,36,42)(23,27,33,39,32,38,45,49)(24,28,35,41,34,40,46,50)(31,37,44,48,43,47,51,52);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,37)(29,39)(30,38)(34,43)(35,44)(36,45)(40,47)(41,48)(42,49)(46,51)(50,52);(7,14)(10,13)(15,20)(17,19)(21,24,25,35,26,34,36,46)(22,28,29,41,30,40,42,50)(23,31,33,44,32,43,45,51)(27,37,39,48,38,47,49,52);(6,11)(10,13)(12,18)(17,19)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(24,34)(28,40)(31,43)(35,46)(37,47)(41,50)(44,51)(48,52)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,d^a=df,c^b=cf,d^b=di,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277568);