#267317

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

915 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 174513);