#262177

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45995	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169373);