#353022

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260218);