#360584

order	512
nilpotency class	3
isoclinism	isoclinic class of #24420
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

2099 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)(5,6,11,15)(7,16,19,30)(8,20,14,10)(9,21,24,28)(12,25,27,18)(13,23,29,22)(17,26,34,36)(31,33,35,32);(5,7,11,19)(6,12,15,27)(8,16,14,30)(9,17,24,34)(10,25,20,18)(13,26,29,36)(21,31,28,35)(22,33,23,32);(5,8,10,6,11,14,20,15)(7,16,18,27,19,30,25,12)(9,21,23,13,24,28,22,29)(17,31,33,36,34,35,32,26);(5,9)(6,13)(7,17)(8,21)(10,23)(11,24)(12,26)(14,28)(15,29)(16,31)(18,33)(19,34)(20,22)(25,32)(27,36)(30,35)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 267780);