#378173

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1431 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=efhi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,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, 285369);