#374395

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1749 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)(7,15)(8,19,21,31)(9,13)(10,14)(11,17)(12,22,28,25)(16,33,20,36)(18,30)(23,29)(24,27)(26,35,32,34);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,20,21,16)(10,24,23,18)(12,26,28,32)(14,30,29,27)(19,33,31,36)(22,34,25,35);(5,8,10,25,9,21,23,22)(6,12,14,31,13,28,29,19)(7,16,18,35,17,20,24,34)(11,26,27,33,15,32,30,36);(1,3,4,2)(6,13)(11,15)(12,28)(14,29)(19,31)(26,32)(27,30)(33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,f^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, 281591);