#374396

order	512
nilpotency class	3
isoclinism	isoclinic class of #196266
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=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, 281592);