#362088

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,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, 269284);