#368143

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1642 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)(19,27)(21,28)(24,29)(25,30)(31,33)(32,36)(34,35);(1,3,4,2)(5,7)(6,12)(8,19)(9,21)(10,20)(11,18)(13,26)(14,27)(15,17)(16,28)(22,31)(23,34)(24,32)(25,33)(29,35)(30,36);(6,13,16,15)(7,18)(12,26,28,17)(14,29,30,22)(19,32)(20,21)(27,35,36,31)(33,34);(5,8)(6,14)(7,19)(9,25)(10,23)(11,24)(12,27)(13,29)(15,22)(16,30)(17,31)(18,32)(20,34)(21,33)(26,35)(28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 275339);