#368869

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1691 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)(7,19)(8,14)(9,26)(10,18)(11,17)(12,16)(13,21)(15,24)(20,31)(22,33)(23,32)(25,36)(27,34)(28,30)(29,35);(1,3,4,2)(5,7)(6,13)(8,24)(9,20)(10,22)(11,23)(12,21)(14,34)(15,31)(16,32)(17,33)(18,19)(25,28)(26,36)(27,29)(30,35);(5,8,10,20,11,25,12,29)(6,14,16,35,17,36,18,31)(7,9,22,28,23,27,21,24)(13,30,32,26,33,15,19,34);(5,9,11,27)(6,15,17,30)(7,20,23,29)(8,22,25,21)(10,28,12,24)(13,31,33,35)(14,19,36,32)(16,34,18,26)

Polycyclic presentation:

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

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, 276065);