#298563

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

998 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)(7,11)(8,16)(9,18)(10,12)(13,19)(14,17)(15,20)(21,22,25,29)(23,30,33,27)(24,34,26,28)(31,36,32,35);(1,2,4,3)(5,7,10,15)(6,11,12,20)(8,17,18,13)(9,19,16,14)(21,23,25,33)(22,27,29,30)(24,32,26,31)(28,35,34,36);(5,8,16,6,10,18,9,12)(7,13,19,11,15,17,14,20)(23,31)(24,26)(27,35)(28,34)(30,36)(32,33);(1,3,4,2)(21,24,25,26)(22,28,29,34)(23,32,33,31)(27,35,30,36)

Polycyclic presentation:

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

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, 205759);