#360581

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 267777);