#270989

order	512
nilpotency class	3
isoclinism	isoclinic class of #270875
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	#36404	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1103 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fi,e^d=eh

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, 178185);