#374048

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281244);