#262276

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

823 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)(5,6,9,14)(7,15,18,11)(8,12,10,13)(16,20,17,19)(21,22)(23,28)(24,26)(25,29)(27,34)(30,35)(31,36)(32,33);(1,2,4,3)(5,7,9,18)(6,11,14,15)(8,17,10,16)(12,20,13,19)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34);(6,12)(7,16)(8,10)(13,14)(17,18)(19,20)(21,24,26,22)(23,30,32,36)(25,34,27,29)(28,35,33,31);(1,3,4,2)(5,8,9,10)(6,13,14,12)(7,17,18,16)(11,19,15,20)(21,25,26,27)(22,29,24,34)(23,31,32,35)(28,36,33,30)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ei,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,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169472);