#238642

order	512
nilpotency class	3
isoclinism	isoclinic class of #233789
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

943 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)(7,8)(9,15)(10,13)(11,14)(12,19)(16,18)(17,20)(21,22,25,29)(23,26,30,35)(24,32,34,28)(27,33,36,31);(1,2)(3,4)(5,7)(6,8)(9,17)(10,12)(11,16)(13,19)(14,18)(15,20)(21,23)(22,26)(24,33)(25,30)(27,28)(29,35)(31,34)(32,36);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,18)(13,16)(14,17)(21,23)(22,27)(24,33)(25,30)(26,32)(28,35)(29,36)(31,34);(7,16)(8,18)(12,17)(19,20)(23,30)(26,35)(27,36)(31,33)

Polycyclic presentation:

2^9|a^2=i,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

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, 145838);