#371188

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi

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, 278384);