#360232

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1534 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)(5,6,10,16,11,17,12,15)(7,18,20,8,13,23,28,22)(9,24,25,14,26,30,27,31)(19,33,34,32,29,21,35,36);(5,7,10,20,11,13,12,28)(6,8,15,18,17,22,16,23)(9,19,25,34,26,29,27,35)(14,21,24,32,31,33,30,36);(2,4)(5,8)(6,13)(7,17)(9,21)(10,18)(11,22)(12,23)(14,29)(15,28)(16,20)(19,31)(24,35)(25,32)(26,33)(27,36)(30,34);(5,9,10,25,11,26,12,27)(6,14,15,24,17,31,16,30)(7,19,20,34,13,29,28,35)(8,21,18,32,22,33,23,36)

Polycyclic presentation:

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

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, 267428);