#298131

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1056 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)(7,13)(8,14)(10,16)(12,17)(15,19)(18,20)(21,22,26,30)(23,27,31,28)(24,32,34,36)(25,35,33,29);(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,23)(22,27)(24,33)(25,34)(26,31)(28,30)(29,32)(35,36);(5,7,14,19,9,15,8,13)(6,10,17,20,11,18,12,16)(21,24,33,23,26,34,25,31)(22,28,35,36,30,27,29,32);(1,3,4,2)(6,11)(10,18)(12,17)(16,20)(23,31)(24,34)(27,28)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,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=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205327);