#110519

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efhi,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^c=eh,f^c=fh

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, 17715);