#377773

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,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, 284969);