#368403

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=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, 275599);