#112101

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

878 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19297);