#110612

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂³	#386
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

812 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17808);