#185302

order	512
nilpotency class	3
isoclinism	isoclinic class of #185249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92498);